Transcript for:
SAT Math Section Lecture Notes

today i'm going to focus on the math section of the sat so i'm going to go over six lessons the first lesson is on algebra solving equations evaluating functions including composite functions and multi-variable functions and then we'll move on to lesson two converting sentences into equations including solving a series of word problems and lesson three ratios proportions probability lesson four averages fractions percentages less than five graphs of linear quadratic and absolute value functions including slope arithmetic and geometric sequences and then lesson six a review of geometry now in each of these lessons i'm going to start out with a review of the important topics equations and concepts that you need to know and then we're going to work on a series of multiple choice problems now for each of these problems i want you to try to solve it first before looking at the solution so make sure you pause the video do the problem first and then check your answer by unpausing the video and watching the solutions this is the best way to get the full benefit from this video if you want to do well on a math section of the sat so let's begin let's start with the first lesson on the algebra part on solving equations evaluating functions factoring and things like that let's go over the basic concepts that you need to know and then we'll work on a few multiple choice problems so let's start with exponents when you multiply common bases you are allowed to add the exponents so x cubed times x to the fourth power is x to the seventh power three plus four seven whenever you raise one exponent to another exponent you can multiply it so five times four is twenty and whenever you divide one base by another common base you can subtract the exponents so nine minus two is seven now let's say if you have a radical the cube root of x to the fifth power how can you convert that into a fractional exponent this is the same as x raised to the five thirds so likewise let's say if you have the seventh root of x to the nine you can rewrite this as x raised to the nine over seven now let's say if you wanted to evaluate four raised to the third power what does that equal now four cubed means that you're multiplying three-fourths four times four times four is 64. likewise if you want to find the cube root of 64 you need to find what number times itself three times is 64. so what times what times what is 64 which we know it to be four so let's say if you want to find the fourth root of 16 what times what times what times what is 16 this is 2 because 2 to the fourth power which is two times two times two times two that's sixteen now let's say if you wanted to simplify a fractional exponent let's say if you have eight raised to the five thirds how can you find the value of this term or of this this number what you could do is you can separate the fraction into two numbers five thirds is the same as one-third times five so eight raised to the five-thirds is the same as the cube root of eight raised to the fifth power eight to the one-third or the cube root of eight is equal to two because two times two times two three times is eight and now we want to find out what two to the fifth power is so two times two times two times two two times two is four this is four and this is two four and four is sixteen so sixteen times two is thirty two so two to the fifth power is 32. let's try another example so let's say if you want to find the value of 16 raised to the 5 4. the first thing you want to do is find the fourth root of 16 and then raise to the fifth power the fourth root of 16 is two and two to the fifth power we know it to be 32 so let's say if you want to solve an equation that looks like this x raised to the two thirds is equal to 16. how would you do it in order to solve for x you need to convert the exponent from two thirds and change it to one the only way you can do that is to raise both sides to the three over two two thirds times three halves is one and so we gotta find out what sixteen raised to the half raised to the third is 16 to the half is the same as the square root of 16 and the square root of 16 is 4 and 4 raised to the third power we know it to be 64. so now let's say if you have a square root the square root of five and you wish to square it what is this equal to the square root of five squared is five the square and the radical cancels because the index number is a two it's always assumed to be a two if there's nothing there you can also see it this way the square root of five squared is basically radical five times radical five five times five is 25 and the square root of 25 is 5. so if you were to see the square root of 8 squared this is simply equal to 8. now let's talk about the absolute value the absolute value of a positive number is a positive number the absolute value of a negative number is also a positive number so with that in mind how would you solve for x in this equation the absolute value of x plus one is equal to three so what's the answer for x whenever you have an absolute value equation you need to write two equations x plus one can equal to positive three and it can also equal to negative three because the absolute value of 3 and negative 3 is the same positive 3. so in the first equation if you subtract by 1 from both sides the first answer is 2 and for the second one negative 3 minus 1 is negative 4. so you get two answers for an absolute value equation so now let's move on to fractions if you want to add five plus two-thirds how can you do so five is the same as five over one whenever you want to add or subtract fractions you need to get a common denominator so the first fraction let's multiply top and bottom by three whatever you do to the top you have to do to the bottom so five times three is fifteen three times one is three so now that we have common denominators we can add the numerators 15 plus 2 is 17 and so the answer is 17 over 3. now what if we wish to multiply uh two fractions if you want to multiply two fractions you multiply across three times five is fifteen and uh two times six is twelve and now we can reduce it let's divide top and bottom by three 15 divided by three is five 12 divided by three is four so the answer is five fourths so when you're multiplying fractions you can multiply across what about if we wish to divide two fractions there's something called keep change flip you can keep the first fraction change division to multiplication and flip the second fraction so 8 times 3 is 24 and 4 times 5 is 20. so let's divide top and bottom by 4 to reduce the fraction 24 divided by 4 is 6 20 divided by 4 is 5 so the final answer is 6 over 5. now let's say if we have a fraction with large numbers now we can multiply across however we're going to get bigger numbers and then we'll have to reduce the fraction so if you're dealing with large numbers simplify first before you multiply so 8 we can rewrite eight as four times two and twenty is four times five fifteen is five times three and twelve is four times three so notice that we can cancel a five and we can cancel a 4 and we can cancel a 3. so right now what we have is 2 divided by 4 which we can reduce that further if we divide both numbers by 2. so then the final answer is one-half so we don't need to know what 8 times 15 is or 20 times 12. so if you're dealing with fractions simplify first before you multiply if you're dealing with large numbers you can do so for small numbers too but for small numbers you don't really need to but for large numbers you can solve faster if you simplify it first so now let's say if you have two fractions separated by an equal sign how can you solve for the value of x whenever you have two fractions separated by an equal sign you can cross multiply so three times four is twelve and five times x plus two is five x plus ten so solving for x let's subtract both sides by ten so two is equal to five x and if we divide both sides by five x is equal to two over five so that's when you can cross multiply so now let's move on to factoring let's say if we have a trinomial where the leading coefficient is one the leading coefficient is the number in front of x squared if you want to factor an expression like this find two numbers that multiply to 15 but that add to the middle term eight so one in 15 multiplies to 15 but as is six i mean sixteen the other option is three and five three times five is fifteen but three plus five is eight and so when we factor it's going to be x plus three times x plus five and because factoring is a common technique that is uh needed on the sat let's do a few examples so go ahead and factor this expression so find two numbers that multiply to 28 but add to 11. so let's make a list 28 divided by 1 is 28 if we divided by 2 is 14 3 doesn't go into 28 but if we divided by 4 is 7 notice that 4 plus 7 is 11. so it's going to be x plus 4 times x plus 7. so go ahead and try this example x squared plus 3x minus 21. so let's make a list of the factors of twenty one so we have one in negative twenty one two doesn't go into it three and negative seven actually this one is not factorable so let's change it let's make it uh x squared minus 4x minus 21 actually plus 4x minus 21. so notice that 3 plus negative 7 is negative four but if we change it to negative three and positive seven it now adds to positive four so this will be x minus three times x plus seven and that's how you would factor it try this one x squared minus 9x plus 20. so what two numbers multiply to 20 but add to negative nine so if they're adding to a negative result um we need two negative numbers if they're going to multiply to a positive number so this could be negative 1 and negative 20 negative 2 and or negative 10 negative 4 and negative 5 but these two add to negative 9 so it's going to be x minus 4 times x minus five but now let's say if the leading coefficient is not one let's say if it's a two what can we do under these circumstances the first thing we need to do is multiply the leading coefficient by the constant term so 2 times negative 2 that does not look like a negative 2 is equal to negative 4. so you need to find two numbers that multiply to negative four but add to the middle term negative three so this has to be negative four and positive one negative four times one is negative four but negative four plus one is negative three so now what we need to do is we need to replace the middle term with negative 4x plus 1x notice that negative 4x plus 1x still adds up to negative 3x so the value of the expression is still the same it's just written in a different way so now at this point what you want to do is you want to factor by grouping so you want to take out the gcf the greatest common factor in the first two terms the gcf is 2x now to find out what's left over on the inside divide 2x squared divided by 2x is x and negative 4x divided by 2x is negative 2. so now if we take out a 1 it's just going to be x minus 2 on the inside if these two factors are the same then you know you're on the right track you haven't made any mistakes thus far so now we're going to take out x minus 2. if we remove x minus 2 from this term the only thing that's left over is 2x and if we remove x minus one from this term the only thing that's left over is plus one and so that's how you factor it so let's try another example where the leading coefficient is not one try this one six x squared plus seven x minus three so if we multiply six and negative three that is equal to negative eighteen so we need to find two numbers that multiply to negative eighteen but add to seven so we have one and negative 18 2 and negative 9 3 and negative 6. notice that 2 plus negative 9 is negative 7 but if we change it negative 2 and positive 9 adds up to positive 7. so let's replace 7x with 9x minus 2x the order doesn't matter we could make it negative 2x and 9x so now let's factor the gcf the gcf for the first two terms is 3x so 6x squared divided by 3x is 2x and 9x divided by 3x is 3. now if we factor out a negative 1 for the last two terms negative 2x divided by negative 1 is 2x negative 3 divided by negative one is plus three so because these two terms are identical we know we are on the right track and what goes in the next fraction is what we see on the outside the three x and a negative one and so that's how you can factor an expression or a trinomial where the leading coefficient is not one so now let's talk about some other functions let's say if you want to factor x squared minus 25 how can you do it right now this is in the form of a difference of perfect squares so a squared minus b squared can be factored into a plus b and a minus b so what you really need to do is take the square root of x squared which is x and the square root of 25 which is 5 one side is going to be positive and the other side is going to be negative so let's say if you want to factor 4x squared minus 81. so what's the square root of 4x squared we know the square root of x squared is x and the square root of 4 is 2 so for 4x squared it's going to be 2x and the square root of 81 is 9. so one is going to be positive and the other is going to be negative now let's say if you want to square factor excuse me 9x squared minus 64 y is to the fourth so what's the square root of 9x squared that's going to be 3x and the square root of 64. we know it's 8 and the square root of y to the fourth what you basically do is take the exponent divided by 2 so it's going to be y squared so it's going to be 8y squared and on one side it's going to be positive and the other side is negative so that's how you can factor using the difference of perfect squares method so now there are some other things that you need to know let's say if you see an equation in this form a squared plus two a b plus b squared this can be factored as a perfect square a plus b squared likewise if you were to see a squared minus two a b plus b squared this is equal to a minus b squared so let's see if you get a question and they give you something like this r squared minus 2 rs plus s squared is equal to 49 what is the value of r minus s notice that r squared minus 2 rs plus s squared that's equal to r plus s squared if you were to factor it so therefore actually not r plus s squared r minus s squared it has to be negative so if you factor it will be r minus s squared so if you want to find r minus s you simply have to take the square root of 49 which is 7. so let's say if you have this equation x squared plus well actually let's say if you have x plus y is equal to 5 and you want to find the value of x squared plus 2xy plus y squared you want to know what that's equal to well if you factor this expression using the equation that we mentioned in the last page this is equal to x plus y squared so basically you just have to square 5. 5 squared is 25 and so you might see a few questions like this on the sat sometimes you have to square it sometimes you have to square root it you just got to know when you have to do what in each case so now let's talk about solving equations that have fractions in it so let's say if you have 5 plus 2 over x and that's equal to 1 how can you solve for x in this equation what i would recommend doing is multiply both sides by the common denominator so there's only one denominator here so let's multiply by x so you got to multiply everything by x 5 times x is 5x 2 over x times x the x variables cancel so you're left over with two and one times x is x so now what we're going to do is we're going to subtract both sides by x and subtract both sides by 2. so these they will disappear 5x minus x is four x and not equal to negative two so if we divide by four we're going to get negative two over four which is negative one half and that's how you could solve for x it's gonna be a lot easier if you multiply both sides by the denominator so let's try another example like that let's say if we have 3 over 2 plus 4 over x and let's say that's equal to 3. so here we have two different denominators 2 and x so let's multiply both sides by the common denominator which is two x so what's three over two times two x which is the same as two x over one notice that the twos cancel and what we have left over is three times x which is 3x and if we multiply 4 over x times 2x the x's will cancel and what's left over is 4 times 2 which is 8. so 2x times 3 over 2 we know it to be 3x the 2's cancel and uh 2x times 4 over x the x's cancel and we have 4 times 2 left over which is 8 and then finally 2x times 3 is 6x so you have to multiply by um you have to multiply 2x by every term if you miss one term your answer will be incorrect it's going to be wrong so make sure you multiply everything in the equation by 2x so now we can solve for x if we subtract both sides by 3x so therefore 8 is equal to 3x and if we divide by 3 x is 8 divided by 3. so now let's move on into functions let's say that f of x is equal to three x plus five if we wish to find the value of f of two how can we evaluate it all we need to do is replace two for x so three times two is six plus five that is equal to 11. so we could say that f of 2 is equal to 11. f of x is equal to y if x is the only variable inside of the function so as you can see 2 is the x value the value on the outside is the y value so using the same function f of x is equal to 3x plus 5. what is the value of x if f of x is equal to 29 how can you find a value of x so we need to realize is that y is equal to 29 so therefore the 3x plus 5 the outside part is equal to 29. and so you set the whole thing equal to 29 and then you can solve for x subtracting both sides by 5 29 minus 5 is 24 and if we divide by 3 24 divided by 3 is 8 so x is eight so keep in mind if the value is on the outside you could set the whole function equal to 29 if it's on the inside you need to plug in for x now sometimes you might have a function that has two variables let's say if you have f of x comma y is equal to x squared plus two y what is the value of f comma three f of three comma five so we could see that x is three and y is five so therefore it's going to be three squared plus two times five three squared is nine two times five is ten so the whole thing has a value of nineteen but now let's say if f of 4 comma y is equal to 28 what is the value of y so notice that in this problem x is equal to 4 and the entire function is equal to 28 so we can say that 28 equals x squared plus 2y and we know that x is 4 so we can plug in 4 for x and let's make some space so therefore 4 squared is 4 times 4 which is 16 and if we subtract both sides by 16 28 minus 16 is 12 and if we divide both sides by 2 12 divided by 2 is 6 so y is 6. so that's how you can solve for a variable if you have a function with two variables so now let's talk about composite functions let's say that f of x is 2x plus 1 and g of x is equal to x squared what is the value of f of g of 2 a composite function is basically two functions where one function is inside the other function in this case g is inside of f so what you should do is start with the the part that's on the inside and then work your way towards the outside so let's evaluate g of 2. so g of x is x squared so g of 2 is 2 squared which is 4. so because g of 2 is equal to 4 i can replace g of 2 with 4. so now i'm looking for f of 4. so i'm going to plug this into the function for f so 2 times 4 plus 1 2 times 4 is 8 a plus 1 is 9 and that's the final answer for this composite function so we've covered a few basic topics that you'll need for the first lesson in this sat course so we've covered factor in solving equations adding subtracting multiplying fractions composite functions and basically most of the stuff that you'll need for the algebra part of the test so at this point let's begin with a few multiple choice problems number one if f of x is equal to three x squared minus five x plus x cubed then f of four is equal to so for all of these problems that we encounter in this video pause the video and try the problem yourself and then see if you can get the answer if you do it that way you're going to get the most out of this video so always try each question before you look at the solution but let's begin if you want to find f of 4 all you need to do is substitute x with four so everywhere we see an x value we're going to replace x with four so now we just have to do some math four squared that's four times four which is sixteen five times four is twenty and four to the third power that's four times four times four four times four is sixteen and sixteen times four is sixty-four 3 times 16 is 48 and 48 minus 20 that's 28 and 28 plus 64 is equal to 92. so therefore d is the correct answer for this problem number two if f of x is equal to x squared plus seven x plus five and f of x is equal to 35 then what is the value of x so what's the first thing that you would do to solve this problem now keep in mind f of x is equal to y so the number that's on the inside of f is equal to the value of x and a number that's on the outside is equal to y so when we see the equation f of x is equal to 35 the number on the outside is equal to y and we're looking for x so we have the equation y is equal to x squared plus seven x plus five and f of x and y are the same thing they equal each other and so now we can replace 35 with y and now we gotta solve for x whenever you see an x squared and an x variable with half the exponent like x to the first power it's a quadratic equation and you may have to solve it either by using the quadratic formula by factoring or even by completing the square so at this point let's subtract both sides by 35. so now we have zero is equal to x squared plus seven x minus thirty so now we need to factor this expression what are two numbers that multiply to negative 30 but that add to seven the two numbers are positive 10 and negative three 10 plus negative 3 is positive 7 but 10 times negative 3 is negative 30. so in this factored form it's going to be x plus 10 times x minus 3. so now what we need to do is set each one equal to zero so if we set the first factor equal to zero x plus ten equals zero we could subtract both sides by 10 so x is equal to negative 10. so that's one possible answer the other answer x minus 3 is equal to 0 if we add 3 to both sides x is equal to positive 3. so out of all the answers that are listed 3 is the right answer answer choice c so that's it for this problem number three if three x plus eight is equal to twenty-four what is the value of seven x plus three so how can we figure this problem how can we find the value of seven x plus three the best way to do this is to solve for x in the first equation and then plug in the value of x in the second expression so let's start with the first equation 3x plus 8 is equal to twenty four so let's subtract eight from both sides so therefore three x is now equal to twenty four minus eight which is sixteen so to solve for x we need to divide both sides by 3. therefore x is equal to 16 over 3. so now we want to find the value of the expression 7x plus 3. so what we need to do is insert the value of x into this expression so right now we have 7 times 16 divided by three plus three to add these two terms we need to get common denominators so we're going to multiply the second term by three over three whatever you do to the top you have to do to the bottom so that the value of the fraction remains equal so 3 is equivalent to 9 divided by 3. now we need to know what 7 times 16 is 7 times sixteen is one twelfth so we have one twelfth over three plus nine over three and if we add those two one twelfth plus nine is one hundred twenty one divided by three so we can see that answer choice a is the right answer for this problem number four if the square root of seven is equal to x minus three then x minus three squared is equal to so how can we solve this particular problem now we could try the approach that we used in the last problem and that is solve for x in the first equation and then plug it in into the expression on the right into x minus 3 squared we could do that and that will work it will give us the right answer but we don't need to if x minus 3 is equal to the square root of 7 then we could square both sides then x minus 3 squared must be equal to the square of square root 7. now square root seven squared is simply equal to seven here's why the square root seven squared is the same as the square root seven times the square root seven this two on top means that we have two square root sevens that are multiplied to each other the square root of seven times the square root of seven is the square root of 49 because seven times seven is 49 and the square root of 49 is 7 7 times 7 is 49 so therefore x minus 3 squared is equal to 7. so b is the right answer now let's just see what would happen if we solve it uh using uh the approach that we used in the last problem so starting with this expression we could solve for x by adding 3 to both sides so 3 plus root 7 is equal to x so now we can try to find the value of x minus 3 squared and so since x is equal to 3 plus the square root of 7 we could take that value and insert it for x so this is going to be 3 plus root 7 minus 3 squared so the 3's will cancel and then what you have left over is root seven squared which we know to be seven so both methods or techniques will work so whichever technique you feel comfortable with that's the one that you should use so b is the right answer for this problem and let's move on to the next one number five if 4x is equal to 12 what is the value of 3x minus 7 squared so let's solve for x so if 4x is equal to 12 we could find the value of x by dividing both sides by 4. so therefore x is equal to 3. so now in order to find the value of 3x minus 7 squared we need to take the value of x and insert it into this expression so it's going to be 3 times 3 minus 7 squared 3 times three is nine and nine minus seven is equal to two and two squared is basically two times two which is equal to four and so therefore a is the right answer for this problem number six if x plus four squared is equal to eight x minus ten squared then the value of x is so how can we figure this problem well let's take the square root of both sides but first let's rewrite the problem so x plus 4 squared is equal to 8x minus 10 squared so we need to take the square root of both sides when you square root a square you need to keep in mind that the index number is two and so the twos will cancel and therefore the square root will get rid of the square and so we don't need the parentheses anymore so what we have is just x plus 4 is equal to 8x minus 10. now what you need to keep in mind is that whenever you take the square root of a number you can get a positive answer and you can get a negative answer so for the negative answer all we need to do is change one side of the equation or multiply one side of the equation by negative one and then it's going to work so let's start with the equation on the left let's subtract x from both sides so we're going to have 4 is equal to 7x minus 10. so next let's add 10 to both sides 10 plus 4 is 14. so 14 equals 7x and then we're going to divide both sides by 7 so 14 divided by 7 is 2. so 2 is one possible answer but notice that it's not one of the choices so therefore we can't really use two as an answer so now let's work with the other equation on the right so first let's distribute the negative to the right side so negative 1 times 8x is negative 8x and negative 1 times negative 10 if we distribute that's going to equal to positive 10. so now let's add 8x to both sides and simultaneously let's subtract both sides by four so this will cancel and that will disappear as well x plus eight x is nine x and ten minus four is equal to six so next we need to divide both sides by nine so therefore we could see that x is equal to 6 over 9 and if you divide both numbers by 3 since they're both divisible by 3 you can get a reduced fraction 6 divided by 3 is 2 9 divided by 3 is 3 so x is therefore equal to 2 over 3 or 2 thirds so b is the right answer for this problem but let's check it let's prove that this value is indeed the right answer so let's plug in two-thirds for x so two-thirds plus four squared should equal to um eight times two-thirds minus ten squared so let's get common denominators four over one is the same as twelve over three if you multiply top and bottom by three you'll get 12 over three and 12 divided by three is four so the value remains the same now eight times two thirds eight times two is sixteen so we have 16 over three and 10 over one what we could do to get common denominators is to multiply ten by three over three so ten is the same as uh thirty over three thirty divided by three is ten so now we can add the two fractions so two thirds plus twelve thirds is equal to uh fourteen over three squared and sixteen minus thirty is negative 14 over 3 squared so on the left side we have 14 over 3 times 14 over 3. that's what 14 over 3 squared means now on the right side we have negative 14 over 3 so that means that we have two negative numbers negative 14 over 3 times negative 14 over 3. in both cases fourteen over three times fourteen over three will be 196 divided by nine and on the right side two negatives um make a positive negative fourteen over three times negative fourteen over three is the same answer in 196 over 9. so therefore because the left side is equal to the right side this equation is true so we can see why b is the correct answer so whenever you take the square root just keep in mind you may have a positive answer and you may have a negative answer so you need to check both to see which one is the right answer or which one is the answer that's listed in this problem number seven if eight times the fourth root of x cubed minus 15 is equal to 49 then the square root of x minus four is equal to so you might see a lot of problems like this on the sat where you have to solve for x in the first equation and then plug in x to the expression on the right side now as you can see the difficulty of these problems are increasing the main idea is the same but the steps that you need to take to solve for x might be different might be longer sometimes it's easier it can vary but just make sure you know your algebra you get you just you gotta know your stuff so let's start with number seven let's rewrite the problem first so the first thing we need to do is we need to add 15 to both sides so 49 plus 15 is equal to 64. next we need to divide both sides by 8. 64 divided by eight is equal to eight now how can we rewrite the radical the fourth root of x to the third how can we rewrite it as a fractional exponent the fourth root of x cubed is equal to x raised to the three-fourths so let me give you another example let's say if you have the seventh root of x to the third this is equivalent to x raised to the three over seven so now how can we solve for x for this equation that we have at this point so we need to change the exponent from three-fourths to one because x is the same as x to the first power or x raised to the one so in order to change it to one we need to raise both sides to the reciprocal of three-fourths which is four over three and whatever you do to the left side you have to do to the right side so when you raise one exponent to another you have to multiply for example x cubed raised to the fifth power is equal to x to the fifteen you multiply three and five so three fourths times four over three the fours cancel and the threes cancel so three-fourths times four-thirds is simply one so we have x raised to the first power is equal to eight raised to the four-thirds so now how can we find a value of eight to the four thirds what you wanna do is you wanna separate the three and the four eight to the four thirds is the same as eight raised to the one-third which is raised to the fourth because one-third times four is four-thirds so the value of this expression is still the same we just rewrote it in a different way so if you want to find out the value of eight to the four thirds the first thing you should do is take the cube root of eight the number on the bottom is the index number that's the root and the number on the top is like the the exponent you're gonna raise it to the fourth power but first let's find the cube root of eight the cube root of eight is a number where before i give you the answer here's what you need to ask yourself if you want to find the cube root of eight find out what number times itself three times is equal to eight so what times what times what is eight the answer is two two times two times two three times is equal to eight so the cube root of eight is two so now we gotta find out what two raised to the four is two to the four is basically two times two times two times two two times two is four and the other two's on the right side is also four so four times four is sixteen so two raised to the fourth power is equal to sixteen and therefore um that's not the final answer yet so we need to avoid the temptation of selecting an answer when we're not finished yet because i was about to do that so what we now have is the value of x x is equal to 16. but our goal is not simply just to find the value of x we want to find a value of this expression the square root of x minus four so let's take the value of x and insert it into this expression so the square root of sixteen minus four the square root of sixteen is four and four minus four is equal to zero so c is the correct answer for this problem number eight if eight minus four over x is equal to x plus four which of the following is a possible value of x so we just got to solve for x in this problem so let's begin what's the first thing that you think that we should do how would you solve for x in this expression now the first thing that i would personally do is i would try to eliminate any fractions before i try to solve for the equation so notice that the denominator of this fraction is x so i'm going to multiply both sides by x so x times 8 is equal to 8x and 4 over x times x is equal to 4 because the x's the x values they cancel so we're just going to get negative 4. and then x times x is x squared and x times 4 is 4x so whatever you do to the left side you have to do the same thing to the right side so we can't just multiply one side by x and not do the same for the other side so this is what we now have notice that we have a quadratic expression we have an x squared and an x so whenever you see that what you want to do at this point is you want to move everything to one side and try to factor the expression use the quadratic equation or complete the square to solve for x at this point so let's subtract both sides by 8x and let's add 4 to both sides so this is zero right here if there's nothing there it's it's a zero so these cancel so on the left side there's nothing left over so it's a zero so zero is equal to x squared and then four minus eight is negative four zero plus four is four so now what we need to do is we need to factor this expression so what number what two numbers multiply to positive four but add to negative four this is negative two and negative two negative two times negative two is equal to positive four but negative two plus negative two is equal to negative four so therefore in its factored form x we have x minus 2 times x minus 2. so if we set x minus 2 equal to 0 and if we add 2 to both sides we could see that x is equal to positive 2. so therefore e is a possible value of x now if you're having difficulty solving for x what you could do is you can plug in each of these answers and see which one is true for the equation so let me illustrate that technique so let's say if you think one is a possible answer you can plug in numbers if you're having difficulty solving for the equation so eight minus four over one we're going to replace x for one is equal to one plus four eight minus four is four and one plus four is five so this is not true the left side does not equal the right side so therefore d cannot be a right um answer so now let's try another value let's try e we know the answer is e so let's replace x with two so four divided by two is two and two plus four is six eight minus two is also six so six equals six the equation is true so therefore we know e is the right answer to this problem so you can always fall back to that technique that is uh basically looking at the answers and plugging it into the equation to see if it works and sometimes that might be the best way to solve the problem it all depends on which technique is faster whichever technique can help you get to the right answer quicker and that's the technique you want to do because the sat is a time test you have to be able to solve the problem very quickly and accurately at the same time number nine if 4x minus 5y is equal to 6 what is the value of 16x squared minus 40xy plus 25y squared so how can we do this problem now don't worry it might look difficult but it's not you need to be familiar with this equation a plus b squared is equal to a squared plus two a b plus b squared so here's the proof a plus b squared is the same as a plus b times a plus b so if you were to foil this expression a times a is a squared and a times b is a b and then b times a is also a b and then b times b is b squared so we can add the two terms in the middle and that will give us uh a b plus a b is two a b squared i mean just two a b so therefore you need to realize that a is 4x in this problem and b is 5y by the way if there's a minus sign it's a minus b squared a minus b squared is a squared minus 2 a b plus b squared but it's very similar so we can see that a is equivalent to 4x and b is equivalent to 5y so therefore if a is 4x that means a squared is 4x times 4x which is 16x squared and if b is 5y that means b squared is 5y times 5y which is 25y squared so then the middle term is 2 times a b so 2 times 4x for a and 5y for b so 4 times 5 is 20 times 2 is 40. so this is negative 40xy so how is this going to help us to get the answer so let's think about what this means so what this means is that 16 x squared minus 40 x y plus 25 y squared is equal to 4x minus 5y squared that's what we know and our goal is to find the value of 16x squared minus 40xy plus 25y squared we want to find out what this what the left side is equal to we don't know right now but we need to use the right side to figure that out now we know that 4x minus 5y is equal to 6. so if that's the case we can replace 4x minus 5y with 6. so therefore the left side is equal to 6 squared which is equal to 36 and that's the answer so it's e you just have to realize that by squaring 4x minus 5y it equals to the value of 16x squared minus 40xy plus 25y squared so therefore all you have to do is square 6 and you'll get the answer so this problem is not hard if you understand it once you understand it getting the answer is easy all you got to do is square 6 and that's it you're done but it's it's the understanding that you need once you understand what to do then math becomes easy number 10 if r squared plus 2 rs plus s squared is equal to 169 what is the possible value of r plus s so notice that r squared plus 2rs plus s squared is in the form a squared plus plus b squared and we know that is equal to a plus b squared so make sure you understand how to factor using this formula because it's going to be very helpful when you're taking your next sat exam so this equation is true therefore we know that r squared plus 2 rs plus s squared is equal to r plus s squared and since r squared plus two rs plus s squared is equal to 169 therefore r plus s squared is also equal to 169 and our goal is to solve for r plus s so what we could do at this point is take the square root of both sides so on the left side we now have r plus s which is what we're looking for and the square root of 16 of 169 excuse me is equal to plus or minus 13. so therefore negative 13 is a possible value of r plus s and that's the answer for this problem so a is the right answer number 11 if the product of x squared minus three x minus ten and three x squared plus two x minus one is zero then x could equal any of the following numbers except so we're looking for the values that x cannot equal so first let's convert the sentence into an equation so the product of x squared minus 3x minus ten product means multiplication and we're gonna multiply this by three x squared plus two x minus one the product of these two terms is equal to zero or these two expressions so we're not going to foil this expression that would be a terrible terrible thing to do what we should do is we need to factor each expression so let's start with the one on the left so what two numbers multiply to negative 10 but add to negative three so let's make a list we have 1 and negative 10 and 2 and negative 5. 2 and negative 5 works 2 times negative 5 is negative 10 but 2 plus negative 5 is equal to negative 3. so this is going to be x plus 2 times x minus 5. and if there's a one in front of x squared once you get the two factors you can simply write it um in this uh in parenthesis if you get these two numbers now for the expression on the right the leading coefficient does not equal one so we're gonna have to factor by grouping so there's gonna be a little bit more work that's involved for that part so i'm going to factor it on the left side first the first thing you need to do is you need to multiply the leading coefficient 3 and the constant term negative 1. 3 times negative 1 is negative three and you need to find two numbers that multiply to negative three but that add to the middle term too so what two numbers multiply to negative three and add to positive two try it so this is none other than positive three and negative one three plus negative one is two three times negative one is negative three so now what we're going to do is we're going to replace the middle term the 2x with positive 3x and negative 1x so that's all we did so far we replaced 2x with 3x minus 1x because 3x minus 1x is still equal to 2x it's simply expressed differently but the value is still the same so now we're going to factor by grouping in the first two terms factor out the gcf the greatest common factor the greatest common factor is 3x you can take out a 3x from 3x squared and 3x that's the most or the greatest that you can factor out when you factor out 3x from 3x squared what's left over to find out what's left over divide 3x squared divided by 3x is x and 3x divided by 3x is 1. so now we're going to factor out negative 1. negative 1x divided by negative 1 is positive x negative one divided by negative one is positive one once you see that these two factors are identical to each other you know you're on the right track so now we're gonna factor out x plus one if we take out x plus one from this term we have three x that's left over and if we remove x plus one from this term we have a negative one that's left over so therefore the expression on the right can be factored to x plus one times three x minus one and all of that is equivalent to zero so now we can solve for x so we can set each factor equal to zero if we set x plus two equal to zero x will equal negative two all you need to do is reverse the sign if x minus five is equal to zero then x is equal to positive five and for this one it's negative one now if three x minus one is equal to zero to solve for x we need to add one to both sides and then we need to divide by three so x is therefore equal to one third so those are the four possible values for x now we're looking for the exception so we could eliminate answer choice a we could eliminate uh b we could eliminate c and we could eliminate d because we have those four answers negative two one third negative one and five so the exception is e x does not equal three um in this equation so therefore e is the right answer for this problem and the factorable expression x squared plus kx plus 24 k is a positive integer which of the following is not a possible value of k so we need to find what value of k will not allow this expression to be factorable so this is like a product sum type problem we need to find two numbers that multiply to 24 but add to k but how can we do that if we don't know what k is so first let's make a list of all the possibilities all of all of the two numbers that multiply to 24. so this would be 1 and 24 2 and 12. 3 and 8 4 and 6. each of these pairs of numbers multiply to 24. 2 times 12 is 24 3 times 8 is 24 4 times 6 is 24. so now what we're going to do is we're going to add each of these numbers because k is the sum 24 is the product the k is the sum 1 plus 24 is 25 12 plus 2 is 14 3 plus 8 is 11 and four plus six is ten so therefore we could eliminate d because k could be equal to ten six times four is twenty four but six plus four is ten and so if k was ten we could factor that expression if k was 11 we could factor it as well and if k is 14 we can factor as well however if k is 7 we can't factor it if we had x squared plus 7x plus 24 this expression is not factorable what two numbers multiply to 24 but add to seven we've already made a list of all the numbers that multiply 24 and that's it if k was 10 we could factor this expression x squared plus 10x plus 24 would be equal to x plus 4 times x plus 6. and so that's why 10 is not the answer and because there's no two numbers that multiply to 24 but add to 7 therefore 7 is not a possible value of k and so answer choice a is the right answer for this problem 13 how can we find the value of x in this expression so what's the first thing that you would do the first thing that we should do is we need to factor each expression so let's start with the expression on the upper left side how can we factor x squared minus 2x minus 24 the first thing we need to do is we need to find two numbers that multiply to negative 24 but that adds to negative two and so this is going to be six and four but which number is going to be negative is it the six or the four it has to be negative six and positive four negative six times positive four is negative 24 and negative six plus four is negative two so we can factor it or write it as x minus six times x plus four now we need to factor this expression as well what two numbers multiply to 12 but adds a positive seven this has to be four and three four times three is twelve four plus three is seven so it's going to be x plus three times x plus four so now we need to factor x squared plus x minus six so what two numbers multiply to six but adds a positive one this is three and negative two so this is going to be x plus three times x minus two so the first thing we need to do is we need to simplify the expression notice that we can cancel x plus four and if we multiply the right side and the left side by x plus three these terms will cancel and the same is true for those terms so what we now have left over on the left side is simply x minus 6 and on the right side 12 divided by x minus 2. so now what we need to do at this point is put this over 1 and cross multiply so 1 times 12 is 12 and we can foil x minus 6 and x minus 2 if we multiply those two so foil in uh x minus six and x minus two is going to be x squared minus two x minus six x and then six times two is twelve so plus twelve so at this point we can combine like terms negative two and negative six is negative eight and now let's subtract both sides by 12. so therefore zero is equal to x squared minus eight x so we're going to factor out an x so zero is equal to x times x minus eight and so therefore x can equal to zero and x can equal to eight any time you see an x on the outside like this x can equal to zero so therefore we're looking for a possible value of x so e is the right answer x could equal 8. 14 4b is equal to 64. then the square root of b times the cube root of 4b is equal to so if 4b is equal to 64 then b is equal to 16 if we divide both sides by 4. 64 divided by 4 16. so now we can find out what the value of this expression is equal to so let's plug in 16 for b the square root of 16 is 4 and 4 times 16 is 64. and the cube root of 64 is a number times a number times a number that equals 64. and that's four four times four times four three times the 64. and four times four is 16. so c is the correct answer for this problem fifteen if x plus y is equal to eight and x minus y is equal to four what is the value of x squared minus y squared so what we need to do is solve for x and y and then we could plug it in to the expression x squared minus y squared to get the answer so let's line up these two equations and notice that we can solve it by using the process of elimination so if we add the two equations x plus x is two x y plus negative y is zero so they cancel and eight plus four is twelve so if we divide both sides by 2 12 divided by 2 is 6. and so now what we can do is we can plug in 6 into the first equation so 6 plus y is equal to 8. subtracting both sides by 6 y is equal to 2. so now we can plug in x and y into this equation so x squared minus y squared that is equal to uh six squared minus two squared six squared is 36 2 squared is 4 and 36 minus 30 minus 4 is equal to 32 and therefore b is the right answer for this problem 16 if 2x plus 3y is equal to 13 and 4x minus 5y is equal to negative 7 then y minus x is equal to so this problem is similar to the last problem so let's use elimination to solve it but let's line up the two equations so two x plus three y is equal to thirteen and uh four x minus five y is equal to negative seven now let's multiply the first equation by negative two so that we can get negative four x and then we can add the two equations so negative four x and then three y times negative 2 is negative 6y 13 times negative 2 is a negative 26. so let's add the first these two equations if we do that 4x and negative 4x will cancel negative 5 and negative 6 if you add them it's negative 11 and negative 7 plus negative 26 is negative 33. if we divide both sides by negative 11 y is equal to positive 3. so now we can solve for x using the first equation so two x plus three y or three times three since y is three is equal to thirteen so three times three is nine and thirteen minus nine if we subtract nine on both sides 13 minus 9 is 4 and then 4 divided by 2 is 2. so we have 2 for x 3 for y so the expression y minus x is therefore equal to three minus two which is equal to one so one is the answer for this problem seventeen if x times y is less than zero which of the following must be true so in this problem let's try to disprove every statement the one that we cannot disprove is the one that must be true so let's choose two values for x and y such that the product is equal to a negative number because the product x y has to be less than zero which means it has to be negative so let's try positive 5 for x and negative four for y so looking at equation one well first this must x y must be less than zero so five times negative four is less than zero because negative twenty is less than zero so five and negative 4 works is true for this equation so now we can test each of the choices to see which one is true so let's focus on x plus y so if x is negative four and y i mean if x is negative five and y is four is it equal to zero negative one does not equal zero so therefore we have disproved number one so that means the answer can't be a and it can't be c so now let's look at uh statement number two three x minus three y is less than zero so three times five minus three times negative four let's see if it's less than zero three times five is fifteen and 3 times 4 is 12. 15 plus 12 is um that's 27 27 is not less than zero 27 is greater than zero so that statement is false so 2 is eliminated because that means it can't be b and it can't be d so the answer has to be e but let's see let's try it just to make sure so x squared plus y squared is greater than zero well x squared will always be a positive number and y squared is always positive unless you plug in zero if you plug in zero for x then it's just zero if you plug in a negative number for x like negative 2 when you square it it's going to be positive so we can see why number 3 is going to be a true statement a positive number plus a positive number is going to be greater than 0. and we can't use 0 for x and y because let's say if we chose zero zero zero times zero is not less than zero it equals zero so we can't use zero for x or for y so which means x and y has to be a number other than zero which means number three will always be true so if we plug in five and negative four we're going to get 25 plus 16 which is greater than zero and let's say if we choose a different test point let's say um x is a a negative number and y is a positive number negative three squared plus five squared will also give us a positive result this is nine plus 25 which is also greater than zero so we cannot disprove number three which means three must be true so therefore e is the right answer eighteen if x is greater than zero which of the following is equivalent to the square root of x to the fifth power so let's go over some rules associated with exponents x squared times x to the third is equal to x raised to the fifth power when you multiply common bases you need to add the exponents and x squared raised to the third is equal to x to the sixth power when you raise one exponent to another exponent you need to multiply the two exponents so now we can answer the question so the square root of x to the fifth is equal to how can you write that as a fractional exponent keep in mind the index number if it's not written it's always assumed to be a two so this is x to the five over two so we need to find out which expression is equal to this one looking at number three x raised to the half raised to the fifth power when you raise one exponent to another exponent you gotta multiply so one half times five is the same as one half times five over one and that's five over two so number three is equivalent to uh this expression so three is true now let's look at number two x to the fourth power times x to the negative three over two when we multiply common bases we need to add the exponents so therefore we need to add four and negative three over two so let's get common denominators let's multiply this fraction on the left by two over two so this is equal to eight over two plus negative three over two eight minus three is five so this expression is equal to x raised to the five over two so number two is a true statement so we can eliminate answer choice a uh c and d because they don't have uh number three which we know to be true so at this point we know two and three is true so we can we can clearly see that b is the answer without even looking at number one now number one is not true let's say if you have x squared plus x cubed this will not equal x to the fifth power you can't add unlike terms so x squared plus x to the one half does not equal x to the to the five over two but let's prove it so let's plug in a test point let's plug in 2 for x actually now 2 let's plug in 4. what is the value of 4 raised to the 5 over 2. this is the same as 4 raised to the half raised to the fifth power so anytime you raise something to the one half it's like finding the square root of that number so the square root of 4 is 2 and two raised to the fifth power that's two times two times two times two times two five times two to the fifth power is thirty two now if we plug in two will we get the same answer or will we get something different 2 squared plus 2 raised to the half 2 squared is 4 and 2 to the half is the square root of 2. the square root of 2 has a decimal value of about 1.4 so this is equal to 5.4 which is not equivalent to 32. therefore number one is not true or it's not equivalent to x raised to the fifth five over two power so therefore e well not e but b is the right answer only statements 2 and 3 are true 19 if c is equal to 4 raised to the x where x and c are both integers which of the following expressions is equivalent to 16 raised to the x plus 4 raised to the x plus 2 power so how can we do this so if we look at our answer choices everything is in terms of c so somehow we need to exchange x for c so we need to do some algebra here let's start with uh the expression 16 raised to the x plus 4 raised to the x plus 2. now we can rewrite 16 as 4 squared in order to convert x into c we need the base 4. so let's convert 16 into base 4. so 16 is 4 squared now last time we went over this property of exponents we said x squared times x cubed is x to the fifth power which is the same as x raised to the 3 plus 2. so we want to do is we want to take an expression in this form and separate it into an expression that looks like what we have on the left so therefore if x raised to the 3 plus 2 power is equal to x squared times x cubed then this expression 4 raised to the x plus 2 power is the same as 4 to the x times 4 squared because when you multiply common bases you can add the exponents so x plus two is the same as or four basically x plus two is the same as four to the x times four squared we can add x and two to get x plus two if we go backwards so you need to understand that property of exponents in order to uh go from this step to this step that we have here so now 4 raised to the 2x is the same as or 4 squared raised to the x is the same as 4 to 2 x whenever you raise one exponent to another you can multiply the two exponents and 4 squared is 16 so what we have is 16 times 4 to the x now instead of writing this as 4 to the 2x i want to write it as 4 raised to the x squared notice that these two expressions are equivalent 2 times x is the same as x times 2 so 4 squared raised to the x is the same as 4 raised to the x squared now the reason why i chose to write it that way is because at this point we can now replace 4 raised to the x with c so this is equal to c squared plus 16 times c so now we're going to factor out c c is the gcf so if we take out c c squared divided by c is c and 16c divided by c is 16. so we get c times c plus 16. so therefore answer choice a is the correct answer for this problem number 20 if the equations above are true which of the following is a possible value of y minus x so let's solve for x if the absolute value of x plus two is equal to seven what is the value of x whenever you have an absolute value equation you can write two equations from it the first equation is x plus two is equal to positive seven and the second equation is x plus two is equal to negative seven the reason why we can do that is because the absolute value of positive seven is positive seven and the absolute value of negative seven is also positive seven and so that's why we can separate into two equations so for the equation on the left if we subtract both sides by two x is equal to positive five and for the second equation if we subtract both sides by two x is equal to negative nine negative seven minus 2 is negative 9. now for the other equation the absolute value of y minus 3 is equal to 4. now let's write two equations y minus 3 is equal to positive 4 and y minus 3 is equal to negative 4. so if we add 3 to both sides 4 plus 3 is 7 and negative 4 plus 3 is negative 1. so these are the possible values of x and y so we need to see what combination will give us one of the answer choices that are listed here so y minus x starting with 7 if we choose 7 for y we can subtract 7 by the x value of 5 which is 2. so 2 is not listed as an answer we can also take the y value of seven and subtracted by the x value of negative nine seven minus negative nine is positive sixteen and that answer is not listed here so now we've used up y so now let's try the other y value so if we use negative one as y we can use positive five for x so negative one minus five is negative six that answer is not there so now if we use negative one for y and then the other x value negative nine negative one minus negative nine is the same as um negative one plus nine which is equal to eight now that answer is listed there so eight is a possible value of y minus x uh using these two equations so as you can see there are four possible values negative 6 2 positive 16 and 8 but only 8 was listed as one of our answer choices so therefore 8 is the answer that we're looking for so e is the right answer 21 if 4c plus b minus a over 7 is equivalent to a what is b in terms of a and c so how can we do this how can we find b in terms of a and c so what we need to do is we need to solve for b that's basically what the question is asking us to do if we can isolate b on one side then a and c will be on the other side of the equation so let's start with the expression 4c plus b minus a over 7 equals a now the first thing i would like to do is get rid of the fraction so i'm going to multiply both sides by 7 so i'm going to multiply every term by seven four c times seven is twenty eight c and b minus a over seven times seven the seventh will cancel and then we'll have just b minus a left over and then a times 7 is 7a so now i'm going to add a to both sides so at this point the a's cancel on the left so what we now have is 28c plus b is equal to 8a so now let's subtract both sides by 28c so then b is equivalent to 8a minus 28c so now at this point we need to factor the gcf what is the greatest common factor between 8 and 28 the greatest common factor is 4 4 can go into 8 and 28. so if we take out a 4 8 a divided by 4 is 2a and negative 28c divided by 4 is negative 7c so therefore we can see that a is the right answer to this problem 22 if f of x is equal to two x plus five and g of x is equal to the absolute value of three minus x plus two what is the value of g of f of one so here we want to evaluate a composite function that's when one function is inside another function so f is inside of g so first let's find out what f of one is equivalent to so that means we need to plug in one into this equation so let's replace x with one so two times one plus five is equal to two plus five which is seven so therefore f of one is equal to seven which means that we can replace f of one with seven so we're looking for g of seven at this point so g of seven is equal to the absolute value of three minus seven plus two three minus seven is equal to negative four and the absolute value of negative four is positive four so four plus two is six so therefore g of f of one is equivalent to six so c is the right answer for this problem twenty three if f of x is equal to the square root of three minus x for all values where x is equal to or less than one and f of x is equal to five minus x squared for all values where x is greater than one what is the sum of f of negative one and f of five so what we really have is a piecewise function the function f of x can be broken into two pieces it can equal the square root of three minus x and it can equal five minus x squared depending on the value of x so the first equation is true when x is equal to or less than one and the second equation should be used when x is greater than one so the goal for this problem is to find the sum of these two function values so let's start with f of negative one if we want to find out the value of f of negative one should we use the first equation or the second equation so when is x equal to negative one in this interval or in this interval we know that it has to be true for the first one because x is less than or equal to negative one so let's plug in negative one into the first equation so it's three minus negative one which is the same as three plus one and that's four and the square root of four is equal to two so now what about f of 5 should we use the first equation or the second equation 5 is greater than 1 so it's in the second equation so it's going to be 5 minus 5 squared if we replace 5 for x so 5 squared is 25 and 5 minus 25 therefore is negative 20. so now we can find a value of f of negative one plus f of five because we're looking for the sum of these two function values f of negative one we know it to be two and f of five is negative twenty so two plus negative twenty is equal to negative eighteen and so therefore b is the right answer for this problem twenty four let f of x comma y be equal to y squared minus 5x so if f of x comma 3 is equivalent to negative 21 what is the value of x so we have a function that contains two variables x and y and so this is equal to y squared minus five x and now we know that f of x comma three is equal to negative 21. so notice that the value of y is equivalent to three so we know that y is equal to three so notice that the left side is equal to the left side of the second equation so therefore the right side of the first equation must equal the right side of the second equation so we're going to set those two equal to each other so y squared minus 5x is equal to negative 21 and we know that y is three so this will allow us to solve for x three squared is equal to nine and we're gonna subtract both sides by nine so negative twenty one minus nine is equal to negative thirty and so we're going to divide both sides by negative 5. so therefore negative 30 divided by negative 5 that is equal to positive 6 and so d is the right answer for this problem because that's all we're looking for we just want to know what is the value of x 25 let the function f of c comma d be equivalent to d squared plus c d minus c squared if f 3 comma e is equal to 61 and e is a positive integer what is the value of e so let's start with the first equation that we have the function is equal to d squared plus c d minus c squared and we know that f three comma e is equal to uh 61. now notice that c is equivalent to three and notice that d is equivalent to e and also the right side of the first equation is equal to the right side of the second equation so we can therefore make the statement that d squared plus c d minus c squared is equal to 61. and we can replace d with e and c with 3. so instead of writing d squared we're going to write e squared plus and then we're going to replace c for 3 so 3 times instead of writing d we're going to write e again and then minus c squared or 3 squared and that's equal to 61. so 3 squared which is 3 times 3 that's equal to 9. so notice that we have a quadratic equation e squared and e to the first power so we might be able to factor it if not we can complete the square or use the quadratic equation but usually these types of problems are factorable because the quadratic equation takes too long and if you're taking the sat you have to do everything fast so let's subtract both sides by 61. so negative 9 minus 61 is equivalent to negative seventy so let's see if this expression is factorable so what two numbers multiply to negative seventy but add to positive three so let's make a list we have negative one and seventy negative 2 and 35 3 doesn't go into 70 and 4 doesn't go into either but 5 goes into it 14 times so negative 5 and positive 14 7 goes into it so negative 7 and 10 and this this works negative 7 times 10 is negative 70 but negative 7 plus 10 is positive 3. so to factor it it's going to be e minus 7 and e plus 10. so let's make some more space so therefore we can write two equations e minus seven is equal to zero and e plus ten is also equal to zero which means e is equal to positive seven and e is equal to negative ten but notice that we have two answer choices negative ten and positive seven which one do we pick now if we go back to the question it said that e is a positive integer so we can't use the negative value so therefore 7 is correct answer choice e is the right answer e is equal to positive 7. 26 let the function h be defined by h of x is equivalent to 7x plus 25. so if the square root of h of b over four is equal to nine what is the value of b so let's start with the inside part of h of b over four notice that there's no multiple choice answers to select so this is a free response problem because typically you'll see some of those questions on the sat so to find h of b over 4 we need to replace x with b over 4. so therefore h of b over 4 is equivalent to this expression so starting with this problem we can replace the h of b over 4 with this expression so what we now have is the square root of 7 times b over 4 plus 25 is equal to 9. so to get rid of the square root symbol we need to square both sides so now 7b over 4 plus 25 is equal to 9 squared and 9 times 9 is 81 so at this point let's go ahead and subtract to both sides by 25 so 81 minus 25 is equal to that should be about 56 but let me make sure my math is correct and yes it's 56 so now we can cross multiply whenever you have two fractions separated by an equal sign you can cross multiply 7b times 1 is 7b and 56 times four that should be 224 so now let's divide both sides by seven so 224 divided by seven is equal to 32 and so that's the answer b has a value of 32. 27 if f of x is equal to this expression what is the value of f of x minus x so f of x minus x is equal to the expression that f of x is equal to that's seven x plus five over three minus four x minus seven over three minus x so keep in mind this portion is equal to f of x so that's all we did we replace f of x with what it equals to uh these two fractions that are subtracted to each other so now let's see if we could simplify the expression on the right side and let's just see what happens so 7x plus 5 over 3 we can separate that into two fractions that's the same as seven x over three plus five over three and we can separate four x minus seven into two fractions by the same time we're gonna distribute this negative sign so it's gonna be negative four x over three and then negative times negative seven that's going to be positive seven over three and then minus x so let's combine these two fractions because they're like terms seven over three minus four over three is basically three x over three and we can combine these two five thirds plus seven thirds five plus seven is twelve so that's twelve over three and then minus x now three x divided by three is simply x and twelve divided by three is four and then we have minus x x minus x is zero so the final answer therefore is four so f of x minus x is equal to four twenty eight if five x is equal to twelve y and y over z is equal to eight over nine then x over z is equal to so how can we do this problem well what we need to do is we need to rearrange some variables so we need an equation that has only x and z so we need to remove y out of the equation so in the first equation let's solve for y so if 5x is equal to 12y if we divide both sides by 12 we're going to get an equation that states that y is equal to 5x divided by 12. now in the second equation y divided by z is equal to eight over nine we can replace y with five x over twelve but before i do that i'm going to rewrite the equation like this y over z is the same as y over one times one over z which is eight over nine so therefore y over one which is basically y we can replace that with five x over twelve times one over z and that's equal to eight over nine it's always better to separate this fraction into two fractions by multiplication rather than plugging this in directly if you plug it in right now it's going to look like this 5x over 12 divided by z and then you're going to have to fix that fraction now you have a complex fraction so i wanted to avoid the formation of a complex fraction and so what i did is i separate this into two fractions by multiplication and it makes it so much easier so now let's continue with what we have at this point so if we multiply 5x and one we we're going to get 5x and then 12 times z is just 12z so right now we have this 5x over 12z is equal to 8 over 9. so i just multiply 5x and 1 and 12 and z so our goal is to isolate x and z we want x on top z on the bottom so we need to get rid of the 5 and 12. so let's multiply both sides by the reciprocal of 5 and 12 which is 12 over 5. so the 12s on the last on the left side excuse me will uh cancel and the fives on the left side will also cancel so therefore we have x over z is equal to 8 times 12 over 9 times 5. now we can multiply 8 and 12 to get 96 and 9 and 5 to get 45 but then we'll have to reduce the fraction it's better if we reduce it now then reduce it later after we get a bigger number so nine is basically three times three the five we can't reduce that further and 4 i mean 12 is 4 times 3 and 8 is basically 4 and 2 but there's nothing to cancel the 4 and 2 so we're going to leave it as 8. notice that we can cancel a 3. so now what we have left over is 8 times 4 which is 32 and 3 times 5 which is 15. and so as you can see it's easier if you reduce the fraction first before you multiply so now you don't have to worry about what 8 times 12 is so if you don't know what 8 times 12 is that's okay you can still get the right answer if you can reduce it first and then multiply later so that's it 32 over 15 is the value of x divided by z 29 if x plus y is equal to 30 and if z over x is equal to 4 and one half z is 20 and x does not equal zero what is the value of x plus z so we got a lot of equations here it might seem like a difficult problem but it's not if you understand it so in order to find the value of x plus c we just need to solve for x and z and then add the two numbers so notice that the first equation has two variables so we can't solve for x or y if there's two variables unless we have another equation and this equation has x and z so we can't use the first two equations because now we have three variables x y and z however if you look at this equation it only has z which means we can solve it so let's start with uh that equation so one half z is equal to 20. so therefore let's multiply both sides by two two times a half is one so one z is equal to 40 which means z is 40. so now let's move on to the second equation the one that has z in it because now that we have the value of z we can solve for x so z divided by x is equal to 4 and we know that z is 40 so we have 40 divided by x is 4. let's cross multiply i'm going to write 4 as 4 over 1 so 40 times 1 is 40 and x times 4 is 4x so if we divide both sides by 4 40 divided by 4 is 10 so x is 10. now notice that we don't need the first equation we don't need the value of y if we wanted to y is 20 if x plus y is 30 and x is 10 10 plus 20 is 30. but our goal is to find the value of x plus z and so we know that x is equal to 10 and z is equal to 40 so 10 plus 40 is 50. and so 50 is the answer for number 29. number 30 if f of x comma y is 2x plus y minus 3 what is the value of f of f comma three comma four comma five so in order to find out the value of this composite function or a function within a function let's start from the inside and let's work our way towards the outside so let's find the value of f comma of f of three comma four so if f of x comma y is equal to two x plus y minus three then we can see that x is equal to 3 and y is equal to 4. so let's plug in 3 for x and 4 for y so this is going to be 2 times 3 plus 4 minus three two times three is six six plus four is ten and ten minus three is seven so therefore f comma f of three comma four is equal to seven so going back to this expression we can now replace f of three comma four with uh seven so we have now is f of seven comma five so we need to find the value of this function now so therefore we can see that x is equal to 7 and y is equal to 5. so in this equation let's replace 7 for x and 5 for y so 2 times 7 is 14 and 14 well let's do 5 minus 3 actually 5 minus 3 is 2 and 14 plus 2 is 16. so the final answer is 16 for this problem so let's start with lesson two we're going to focus on the ability of converting a sentence into an equation so let's go over a few concepts and then we'll work on some multiple choice problems so let's start with number one five more than twice the value of y how would you write an equation from that sentence so five more five plus and then twice the value of y that's 2y and that's it number two the sum of five times the number and the square of the number is eight so five times the number let's call the number x so 5x and the square of a number which is x squared and since we have the word sum it's going to be plus and then is is the same as equal to so is 8. sally is one year less than three times as old as john so sally is when you hear like less than and after like two then three times if you see it like that the less than part comes after not before so sally is three times as old as john less one it's kind of backwards cara is three times the difference between the ages of jeremiah and susan let's start with the difference between jeremiah and susan so that's j minus s and then three times the difference so three times j minus s and that's equal to uh cara's age the sum of two numbers is eight let's say the two numbers is x and y sum means addition and then the product which means multiplication is five so x y equals five now number six the sum of half a number and twice another number is less than or equal to nine so let's say the two numbers are x and y so when we hear the word sum we're thinking of addition half a number let's say half of x plus twice another number two times y is which is usually an equal sign but it's less than or equal to so we're dealing with an inequality less than or equal to 9. so i just want to give you a little warm up of how to convert sentences into equations so make sure you develop this ability as best as you can because to do well in the sat at least the math part you need to be able to convert sentences into equations but now let's go over some other concepts that you need to be familiar with the first thing is averages the average of a number is the total value of a number divided by the number of values so let's say if you want to find the average between 10 12 14 16 and 18. you will add these numbers up and then simply divide by five now sometimes you may need to know what the total value is the total value is equal to the average times n let's calculate the average 10 plus 12 fourteen plus sixteen plus eighteen divided by five is fourteen so the average is fourteen also five times fourteen is seventy and seventy represents the sum of the five numbers so now let's move on to consecutive integers when you hear the word consecutive what do you think of an example of consecutive integers is a number that occurs right after another number so consecutive positive integers would be like seven eight nine ten consecutive negative integers would be like negative five negative four negative three and they need to know odd numbers and even numbers even numbers are like 2 4 6 8 and so forth odd numbers are like 1 3 5. now if you hear the word consecutive even integers that would be like 2 4 6 8 10 and so forth those are consecutive even integers now you need to know the difference between whole numbers natural numbers and integers an integer could be negative it can be positive or it can be zero these are considered integers a whole number includes zero and positive integers natural numbers do not include zero but they do include positive integers so just in case you see these terms on the exam you know what they mean now when you hear the word multiple what are multiples of seven multiples of seven are 7 14 21 28 35 and so forth just in case you see these words in a sentence you need to understand how to turn them into an equation so you have to know what these words mean next in our list is the terms inclusive and exclusive so let's say if you want to make a list of all the consecutive odd integers between 5 and 12 inclusive so that would include 5 7 9 and 11 those are the odd integers and they're listed consecutively and 12 is not odd so it's not included if it was exclusive that means it doesn't include 5 or 12. so exclusive would be 7 9 and 11. so if you want to find all of the odd integers between five to twelve exclusive not including five and twelve it's seven nine and eleven so list all the integers between one to seven inclusive and exclusive so inclusive that means including one and seven so it's one two three four five six and seven exclusive you're not including one and seven you're excluding them out of the list so it's going to be two three four five and six so now you know what these terms mean the last thing you need to be familiar for the next few multiple choice problems that you're going to go into is the equation for distance rate and time you've seen this equation many times d equals rt d represents distance r represents the rate which is usually speed and t represents the time so let's say if you have a car going at 30 miles per hour what distance will it travel in four hours if a car is moving at 30 miles per hour what that means is that in one hour it's going to travel a distance of 30 miles so in four hours it's going to cover a distance of 120 miles and that's the idea between distance rate and time by the way make sure the units match so if you have if your rate is in miles per hour the time has to be in hours and so if the rate is in miles per hour the distance have to be in mouse so the units have to match if they don't match make sure you convert one unit into the appropriate unit that's going to work in this equation all right so that's basically it so let's uh jump into some multiple choice questions and uh let's get started 31 if the average of x z and 70 is 10 more than the average of y z and 30 what is the value of x y so the equation for the average is equal to the total or the total value divided by the number of values so let's say if we have three numbers 10 11 and 12. these numbers are consecutive and if we wanted to find the average it would be 10 plus 11 plus 12 divided by 3 which is equal to the number in the middle which is 11. so that's how you find the average of a number but now let's see if we can use that to solve this problem so if the average of x z and 70 the average of those three numbers is the sum x plus z plus 70 like we did 10 plus 11 plus 12 and then because we have three numbers the average is going to be the sum divided by 3. so x z and 70 is is is equivalent to equal is 10 more than the average of y plus z plus 30 divided by 3. so that's the equation that we have now there are three variables x z and y and our goal is to find x minus y we can't isolate and solve each variable however we could possibly get x and y by itself and that's what we have to try to do here because if we have three variables you can't solve for each variable unless you have three equations and we only have one equation so let's multiply everything by three to get rid of the fractions so the fraction on the left times three the threes will cancel and so we're gonna get x plus z plus 70 left over and then we're going to multiply the 3 by 10 and so that's going to equal 30 and then the three times this fraction the threes will cancel and so that will equal y plus z plus 30. so at this point we can add like terms and at the same time we can subtract both sides by z so the z variables will cancel and 30 plus 30 equals 60. so right now we have x plus 70 is equal to 60 plus y so if we subtract both sides by 60 what we now have is x plus 10 is equal to y let's move the 10 back to this side so x equals y minus 10. and now let's subtract both sides by y so then we now have is x minus y is equal to 10. well negative 10. we can't forget about this negative sign and so that's it that is the value of x minus y so b is the right answer for this problem 32 if five more than three times the number is 15 less than that number what is the number so let's convert this question into an equation so we have five more more represents plus five more than three times the number we're going to call the number x is is the same as equals 15 less than that number so it's the number minus 15. what is the number so we just got to solve for x let's add 15 to both sides so 20 plus 3x is equal to x and so now let's subtract by 3x so x minus 3x is negative 2x and if we divide both sides by negative 2 we can see that x is equal to negative 10. so therefore the number is negative 10 so a is the right answer 33 the sum of three consecutive positive even integers is z in terms of z what is the sum of the first and second integers so 5 6 and 7 are integers consecutive integers they're positive but they're not even 5 is odd 6 is even but numbers like 8 10 and 12 they're even integers and they're consecutive so this is the pattern of numbers that we're looking for so then the sum of three consecutive positive even integers is d in terms of z how can we find the sum of the first and second integers feel free to pause the video and try this yourself so let's say that the first integer is x therefore the second one will have to be x plus two the second one is two more than the first one and the third one is going to be x plus four so let's say if x was uh eight eight plus two would be ten and eight plus four is twelve so as you can see these are consecutive positive even integers now the question the question stated that the sum of these three consecutive even integers is equal to z so that's the equation that we have and somehow with this equation we need to find out what the sum of the first and second integers are in terms of z so here's the first one this is the second and this is the third so what we're going to do is we're going to solve for x in terms of z first so x plus x plus x is three x and two plus four is six so three x plus six is equal to z and if we subtract both sides by 6 3x is equal to z minus 6 and if we divide everything by 3 x is equal to z minus 6 over 3. so now let's save that answer so now our goal is to find the sum of the first and the second integer so the first integer we know it's x but in terms of z we know that x is equivalent to z minus six over three so let's move this somewhere else and the second one is x plus four so z minus six over three plus two so this is for the second integer so now we can add these two terms together z plus z is 2 z and negative 6 plus negative 6 is negative 12 divided by three now we need to do something with the two so let's write it as two over one and let's try to get common denominators so we're gonna multiply the two by three over three so then this is going to be plus 6 divided by 3. now we can now add the numerators so it's going to be 2z and then negative 12 plus 6 is minus 6 divided by 3. so this is the sum of the first and second integers in terms of z so therefore d is the right answer to this problem by the way the video that you're currently watching is a two hour trailer version of a longer six hour video so this video currently has the first lesson and part of the second lesson but if you want access to all six lessons i'm gonna post a link and you can check out that eight hour video when you get a chance so let's continue working on the next problem 34 if the remainder is 5 when a positive integer b is divided by 7 then what is the remainder when nine b is divided by seven so before we solve uh this problem let's go over an example uh situation so let's say if we wanted to divide 37 by seven so if we were to use long division we can see that 37 goes into i mean 7 goes into 37 five times and we can see that two is a remainder so in fraction form we know that seven goes into 37 five times and the remainder is two and since we couldn't divide it by seven we leave it as 2 over 7. now to see why this works 37 is the sum of 35 plus 2. notice that the left side is equal to the right side so 37 over seven is 35 over seven plus two over seven and thirty five over seven is five and then the part that we can't reduce we leave it as two over seven so now let's go over another example let's say if we wanted to divide 9 into 49 or divide 49 by 9 9 goes into 49 five times and since 5 times 9 is 45 the remainder is 4 but we write it as 4 over 9 because um that's the part that we couldn't because we tried to divide 4 by 9 but we couldn't so we'll leave it as four over nine so now let's see if we can apply this situation to this problem but i wanted you to understand uh the process of dividing two numbers and getting the remainder so when a positive integer b is divided by 7 the remainder which is this number the remainder is 5 but we have to write it as 5 over 7. now we don't know how many times uh 7 can go into b so we're going to say that the amount of times that 7 goes into b we'll call it just n now our goal is to find out what is the remainder when nine b is divided by seven and that's what we want to do so if we compare b over seven to nine b over seven basically it's simply nine times its value so let's multiply both sides of this equation by nine so on the left we're going to get 9b over 7 and we've got to multiply everything by 9. so n times 9 is 9n and then 5 times 9 is 45 over 7. so now notice that let's focus on this part forty five over seven we can reduce that seven can go into forty five the question is how many times seven can go into 45 six times and since 7 times 6 is 42 45 minus 42 is 3 so 3 is the remainder so we can rewrite this as 9b over 7. 7 goes into 9b at least 15 times well or 15n where n is the number of times it can go into well maybe that's not really accurate i should say nine n plus six there we go okay that's more accurate so seven goes into nine b nine n plus six times and the three is remaining so all i did was i replaced the 45 over 7 with 6 plus 3 over 7. so that's why it's 9n plus 6 plus three over seven and so the remainder is three so three is the final answer to this problem so let's see if we can prove it think of a number in which seven could go into but the remainder is five so let's use 19. so we're going to say b is 19. so seven goes into nineteen two times and seven times two is fourteen and nineteen minus fourteen is five so here we got a remainder of five so now if we multiply 19 by nine will the remainder be three so nineteen times nine is a hundred and seventy one so how many times does seven go into one seventy one seven times twenty four is 168. so 7 can go into 171 at least 24 times and 171 minus 168 is 3. so we could have re written this at like this 171 is basically 168 over 7 plus 3 over 7. 168 plus 3 is 171 and 168 divided by 7 is 24. so we get 24 and 3 7. basically you can convert this into a mixed number if you want but as you can see the remainder is 3. so therefore c is the right answer for this problem 35 a basketball team won 11 more games than it lost if the team played a total of 81 games and there were no ties how many games did the team lose so for this one we want to convert the sentences into equations so let's start with this sentence or this portion of the sentence the team played a total of 81 games so some games they won and sometimes they lost so w plus l the wins and the losses should add up to 81 games since that's the total games that they played now let's focus on the first sentence let's see if we can turn it into an equation the team won 11 more games than it lost so that means that w is equal to l plus 11. the number of wins is 11 more than the number of games that the team lost so that's the equation for the first sentence at this point we have two equations and two variables we can solve using the method of substitution or elimination since we have w on one side in the second equation substitution is the best option so let's replace w in the first equation with 11 plus l in the second equation since they equal each other so what we now have is 11 plus l plus l is 81. so therefore 2l equals or 2l plus 11 is equal to 81. so if we subtract both sides by 11 2l is equal to 81 minus 11 which is 70. and if we divide both sides by 2 70 divided by 2 is equal to 35 so that's how many games the team lost they lost a total of 35 games so c is the right answer if you want to find how many games were one you can use the first equation so w plus l is 81 so w plus 35 is equal to 81 and 81 minus 35 that's about 46. so notice that 35 plus 46 adds up to 81 and 46 is 11 more than 35 so this is the answer for w and l because at those values equation one and equation two are true but c is the right answer since we're looking for the number of games that the team lost 36 when 4 times the difference of a number n and 15 is divided by 3 the result is 12. what is the value of n so let's turn the sentence into an equation so four times the difference of a number n and fifteen let's focus on that part the difference of a number n and fifteen the difference between n and fifteen is simply n minus -15 and it says 4 times 4 times the difference so it will be 4 times n minus 15 in parenthesis and this is divided by 3 and when it's divided by 3 the result is 12. so the result means equal so now our goal is to solve for n so we can write 12 as 12 over 1 and since we have two fractions separated by an equal sign we can cross multiply so 3 times 12 is 36 and 1 times 4n minus 15 is 4n minus 60 if you distribute the 4. 4 times negative 15 is negative 60. so now let's add 60 to both sides so 36 plus 60 is equivalent to 96 and now we need to divide both sides by four so 96 divided by four is 24 and that is the value of n so e is the right answer a certain sample of bacteria triples in number every hour if there were eight bacteria in the sample at the start of the experiment how many bacteria were there after six hours so initially there was eight and after the first hour there's gonna be eight times three which is 24 and after the second hour times three and then the third hour and then the fourth and then the fifth and then the six every hour it triples so after six hours is going to be eight times three raised to the sixth power now three to the sixth power is about 729 so eight times 729 is equal to 5832 so that's going to be the amount of bacteria after six hours so c is the right answer 38 three bananas and eight grapes cost a dollar ninety one 14 bananas and 25 grapes cost eight dollars and five cents what is the cost of eight bananas and sixteen grapes so we need to write two equations and solve for the number of bananas and grapes then we can find the cost of 8 bananas and 16 grapes so let's begin let's start with the first sentence so three bananas or three times b plus eight grapes eight times g has a cost or is equal to 191 now for the second equation 14 bananas and 25 grapes costs eight dollars and five cents so these are the two equations so now we need to solve for either b or g first so let's start with let's solve for b let's multiply the first equation by negative 14. so and then the second equation we're going to multiply by positive 3. so negative fourteen times three b is negative forty two b and eight times negative fourteen that's going to be negative one twelve g and 1.91 times negative 14 is about negative 26.74 now for the second equation 14b times 3 that's positive 42b 25 times 3 is 75 g and 8.05 times 3 that's going to be positive 24.15 so if we add the two equations these two variables cancel and negative 112 g plus 75 g that's equal to negative 37 g and negative 26.74 plus 24.15 that's negative 2.59 so now we can solve for g so if we divide both sides by negative 37 g is going to be let's see negative 2.59 divided by negative 37 is equal to 7 cents so that's the cost of one grape now using this equation let's find the cost of a banana so let's solve for b three b plus eight times the value of a grape which is seven cents is equal to a dollar and ninety one cents so eight times seven cents that's gonna be 56 cents and a dollar 91 minus 56 cents that's gonna be a dollar thirty five for three bananas so therefore if we divide it by three we can get the cost of a single banana so each banana costs 45 cents so now at this point we could find the value of 8 bananas and 13 and 16 grapes so 8b plus 16g so let's plug in 45 cents for b and seven cents for g so eight times forty five cents that's going to be three dollars and sixty cents and sixteen times seven cents it's about a buck twelve so a dollar twelve plus three sixty that is equal to a total value of four dollars and seventy two cents so that's the cost of eight bananas and 16 grapes which is between the cost of three bananas eight grapes and 14 bananas 25 grapes so b is the right answer 39 if b is an integer that satisfies the inequality above what is the sum of the largest possible value of b and the smallest possible value of b so to get the smallest possible value of b we can use this equation four is less than the square root of b and to find the largest possible value of b we can solve this equation the square root of b is at most eight so therefore let's square both sides for the first equation so 4 squared is 16 and the square root of b but squared is simply b and if we square the other side b is less than or equal to 8 squared 8 times a to 64. so therefore what is the largest value of b if b is less than or equal to 64 then the largest that b can equal is 64. now if b is greater than 16 what is the largest possible value of b it is not 16 b has to be greater than 16 and b is an integer so it can't be like 16.1 so b has to be greater than or equal to 17 because it has to be an integer like 16 17 18 but since it's greater than 16 it can't equal 16 but it can equal 17. so our goal is to find the sum of the largest possible value of b and the smallest possible value of v excuse me b so the symbolis value of b is 17 and the largest value of b is 64. 17 plus 64 is equal to 81 and therefore e is the right answer number 40 bonnie is five years younger than roger and four times as old as dana if dana is d years old how old is roger in terms of d so let's start with the first part of the sentence bonnie is five years younger than roger how can you write an equation between bonnie and roger so therefore bonnie is is roger's age but minus five so if roger's like 40 bonnie would be 35 40 minus 5 is 35 so therefore bonnie would be 5 years younger than roger if bonnie was 35 and roger was 40. now bonnie is four times as old as dana so b equals four d so if dana's like eight four times a is thirty two bonnie will be thirty two so now if dana's d years old how old is roger in terms of d or dana so what we need to do is get an equation and solve for r in terms of d so notice that b is equal to r minus five and b equals four d so since we know that b equals b we can replace b with r minus five on the left side and on the right side we can replace b with four d so now we can solve for r so if we add five to both sides r is equal to four d plus five so roger is four times as old as dana plus five so let's say if dana is 10 years old that means bonnie is four times her age so bonnie is 40. and roger is four times dana age plus five so 40 plus five rogers 45 and he's five years older than bonnie or bonnie is five years younger than roger so we can see how the numbers work out here but the answer that we're looking for is how old roger is in terms of d so roger is four d plus five so therefore a is the right answer to this problem