welcome to another video from Ultimate algebra.com in this video we will be looking at how to answer Algebra 1 questions the easiest way please this video will not be exhaustive for a complete course deeper dive with more examples please check out our ultimate algebra course at ultimate algebra.com let's Dive Right In question 1 x + 2 = 5 solve for x the first thing we are looking at is how to solve one-step equations you have to be good at solving one-step equations in order to pass any math test the idea of solving equations is to move everything with the X to the other side of the equation by performing the opposite operation on it we want only the X to be on one side of the equation so here we want to move the plus two to the other side of the equation we do that by performing the opposite operation on both sides of the equation the opposite operation of addition is subtraction the opposite operation of multiplication is division the opposite operation of exponent is the root or radical we know the opposite operation of addition is subtraction so we will subtract two from both sides once the opposite operation is done those two numbers simply cancels out so here the two will cancel out 5 - 2 is 3 so X will be equal to 3 we have a free complete video on solving equations with a lot more examples please check it out with the link in the description for more let's move on to question two question 2 2x + 3 = 11 solve for x here we are looking at solving 2ep equations we said earlier that the whole idea of solving equations is to get get rid of everything and leave the X on one side of the equation for this question we will see that we have to get rid of the multiplication by 2 in the plus three 2x is the same as 2 * X when there are more than one operations we use the idea of the reversal of the order of operations to know which one to get rid of first so here is the order of operations if you are not familiar with the order of operations please revise our videos on it it's very important knowing the order of operations will make solving of equations super easy we can see that in the reversal of the order of operations that's from bottom to top we have addition before multiplication so we will get rid of the plus three first we get rid of the plus three by performing the opposite operation on it so we will subtract three from both sides the three will cancel out 11 - 3 is 8 so we have 2 x x = 8 next we will get rid of the multiplication by two by dividing both sides by two since division is the opposite of multiplication the two will cancel out 8 / 2 is 4 therefore x = 4 question three 3x^2 + 8 = 20 solve for x we are looking at solving multi-step equations the process of solving is just like solving twostep equations we want to get the X on one side of the equation in order to do that we have to get rid of the multiplication by three the exponent 2 and the Plus 8 we will use the reversal of the order of operation to know which one to perform first let's bring our order of operations so here is our order of operations in the reversal we will notice that we have to to do the Plus 8 first then we will do the multiplication by three and then we do the exponent two we will get rid of the Plus 8 by performing the opposite operation on it subtract eight from both sides the eight will cancel out 20 - 8 is 12 so now we have 3x^2 = 12 next we have to get rid of the multiplication by three we do the opposite operation we divide both sides by three the 3 will cancel out 12 / 3 is 4 so now we have x^2 = 4 we finally have to get rid of the exponent two we do the opposite operation the opposite operation of squared is square root we find the square root of both sides this will cancel out the square root of 4 is 2 therefore x = 2 question 4 4x + 5 = 9 + 2x solve for x in our previous questions we have been having the X represented only once example 4x + 5 equals 9 but here we have the X represented twice in a case like this we want to move the X values to one side of the equation and work on it so here you can choose to move the 4X or 2X I'll move the 2x to the other side of the equation to do that since the 2x is adding you will subtract 2x from both sides of the equation the 2x will cancel out now 4x - 2x is 2x so we have 2X + 5 = 9 we now have a familiar two-step equation which you should be able to solve if you've watched the previous questions but let's go over it we want to get rid of all the numbers at attached to the X so we can have the X alone on one side of the equation to achieve this we know that we must get rid of the time 2 in the + 5 let's bring our order of operations please you don't need to be writing the order of operations in your Solutions we are using it for teaching purpose we are using the reversal of the order of operations so we are working from the bottom up we will see that in this form we must do the addition first before the multiplication to get rid of the plus 5 we must perform the opposite operation on both sides of the equations so we will subtract five from both sides the five cancels out 9 - 5 is 4 we now have 2x = 4 next we will get rid of the multiplication by two we do this by performing the opposite operation on both sides of the equation so we will divide both sides by two the two will cancel out 4 / 2 is 2 therefore x = 2 question five the absolute value of x + 3 = 7 find X for absolute value equations we equate the absolute value to the positive and negative of what is on the other side of the equation here we will equate the absolute value to positive and negative of the 7 so we have x + 3 = 7 and x + 3 = -7 we solve both equations for the first one we subtract three from both sides the three will cancel out 7 - 3 is 4 for the second one we subtract three from both sides the three will cancel out -7 - 3 is -10 so our answer is x = 7 and and X = -10 question 6 the absolute value of x + 1 + 6 = 9 find X in the previous question everything on the left side was in the absolute value Marks here the trick is that the plus 6 is not in the absolute value you need to remove everything that is not in the absolute value Mark and get only the absolute value on one side for you can equate to the negative and positive so we will first start by subtracting six from both sides the six will cancel out 9 - 6 is 3 now we have the absolute value of x + 1 = 3 we have only the absolute value on one side of the equation so we can equate x + 1 to the positive and negative of what is here as usual we have x + 1 = -3 and x + 1 = POS 3 we subtract one from all sides the one will cancel out for this one -3 - 1 will be -4 so x = -4 for this one 3 - 1 will be 2 so x = 2 so x = -4 or x = 2 question 7 s < TK x + 3 - 2 = 1 find X we are looking at radical equations a radical equation is an equation in which the variable is contained inside a radical or root sign here we see that the x is under the square root sign similar to what we did for absolute value equations we want the isolate the radical on one side of the equations we add two to both sides the two will cancel out 1 + 2 will give us three now we have the square < TK of x + 3 = 3 next we want to eliminate the square root by squaring both sides of the equation for the left side the square root will cancel out the square we just get x + 3 3^ 2 is 9 we have a one-step equation we subtract three from both sides the three will cancel out 9 - 3 is 6 therefore x = 6 question 8 4 / the quantity x - 5 = 3 / X find X we are looking at rational equations for rational equations we have the x or variable in the denominator the first step is to remove the fractions typically we will use the least common denominator method but for this question we can just cross multiply 4 * X is 4 x 3 * x - 5 is 3 x - 15 we expanded it 3 * X is 3x and 3 * -5 is -15 next we want to isolate the X on one side of the equation we subtract 3x from both sides the 3x will cancel out 4x - 3x is 1 X or simply X so we have X = -5 as our answer question nine solve for x given that yal mx + b here we are looking at change of subject or transposing of formula this is the slope intercept form of the equation of a line when you are given any formula you should be able to find any of the variables this is nothing different from what we've been doing so far to solve for x we we want to have the X on one side of the equation and everything else on the other side of the equation to do that we have to move the times M and the plus b we will use the reversal of the order of operation to know which one to perform first let's bring our order of operations in the reversal we will notice that we have to do the plus b first then we will do the multiplication by m we will get rid of the plus b by performing the opposite operation on it subtract B from both sides the B will cancel out you cannot subtract y minus B because they are dissimilar as we learned in addition and subtractions in algebra so we will have y - B = MX next we have to get rid of the multiplication by m we do the opposite operation we divide both sides by m the m will cancel out there's nothing we can reduce on this other side so x = y - B all over M question 10 -3x + 1 is greater than 7 solve for x here we are looking at solving inequalities the process is the same as solving equations there is a slight difference when you multiply or divide by a negative we want to get rid of everything and leave it X on one side of the inequality sign for this question we will see that we have to get rid of the multiplication by -3 and the plus one let's bring our order of operations we can see that in the reversal of the order of operations that's from bottom to top we have addition before multiplication so we will get rid of the plus first we get rid of the plus one by performing the opposite operation on it so we will subtract one from both sides the one will cancel out 7 - 1 is 6 so we have -3x is greater than 6 next we have to divide both sides by the -3 so we can get the X by itself in inequalities when you divide or multiply by negative the inequality changes so here the greater than becomes less than please take note of this nearly all wrong answers are because of this mistake now the -3 will cancel out 6 / -3 is -2 therefore X is less than -2 question 11 solve the inequality -3 less than x + 8 less than 20 here we are looking at combined inequalities this question is the same as -3 less than x + 8 and x + 8 less than 20 we just com combine them the solution is exactly the same instead of having two sides you now have three sides to get the X by itself we have to subtract eight from all three sides the8 will cancel out here -3 - 8 is -11 then 20 - 8 is 12 so our answer is -1 less than x less than 12 question 12 graph the inequality X greater than -4 here we are looking at graphing inequalities let's bring our number line when graphing inequalities the first thing is to locate your point which will be the number here it is -4 then you'll draw a shaded or unshaded circle at the point if you have less than or greater than then the circle will not be shaded if you have less than or equal to or greater than or equal to you will use the Shaded Circle so basically if it has an equal to you will shade in this case since it's just greater than we will not shade the circle then finally we draw an arrow the easiest way to get this always right without thinking is to make sure your inequality is in the form variable inequality sign and then the number in that form the direction of your arrow will be the same as the direction of the inequality sign here since this is in that form we will draw our Arrow facing here and we are done question 13 in order to ship 2,500 gallons of a product to another country Stan shipping company had to package the gallons into boxes if they had 20 package boxes and 100 gallons left that are not in boxes how many gallons were in a box this kind of two-step equation word problem is is very common we are first going to solve it in details for teaching purpose then I'll show you how you can solve it in less than 10 seconds on an actual test you have three values in these type of questions let's write them down we have 2,500 gallons we have 20 boxes and finally we have 100 gallons the 2,500 gallons represents the total so we have equal 2,500 the 20 box is what I call the group The gallons have been grouped into boxes for most questions the group can also be identified as the number that represents something different from the other two numbers so here 2,500 represents gallons the 100 also represent gallons but the 20 represent boxes so the 20 will be the group the group is the one with the X so we will have 20x we can now add the 100 gallons left to to the equation and solve for the X in this two-step equation subtract 100 from both sides these will cancel out 2500 - 100 will be 2,400 we now have 20x = 2,400 divide both sides by 20 the 20 will cancel out 2,400 ided 20 will be 120 this means there were 120 gallons in each box the hard part of this question is to be able to pull out the values from the word problem we went through a detailed solution for teaching purpose let's look at how you can speed up solving this question first there's absolutely no reason to write this part if you know what you're doing you can go straight to writing your two-step equations we have our group 20x our other + and our total 2,500 then you can solve the two-step equation equation an even faster method to solve this question in less than 10 seconds is to do 2,500 minus 100 divided by 20 on your calculator to get 120 we did the total minus other divided by the group a big caution when using fast methods is to note that they are very specific to specific questions and little twists to the question can let you get it wrong so when in doubt use the longer methods question 14 five added to Thrice Michael's age is 50 how old is Michael for question like this it is good practice to start with identifying your unknown value and representing it by a letter let's say x the unknown value is usually what the question is asking you to find here it is Michael's age let's represent it with X now we just translate we know added to is addition Thrice is three times so Thrice Michael's age is 3x and is means equal to so this is 5 + 3x = 50 you can now solve the two-step equation we subtract five from both sides the five will cancel out 15 - 5 will be 45 so we have 3x = 45 next we will divide both sides by three this will cancel out 4 5 / 3 is 15 therefore x = 15 Michael's age is 15 years question 15 which of the relations below is not a function to answer this question let's go through a few things about functions when you have a notation example 25 the two represents the input value that's x value the five represents the output value that's the Y value this is the same as what we will learn in graphing points for a relation to be a function no input value can have multiple output values also all input values must have output values let's look at examples here the input values are these and the output values are these note that each input value corresponds to only one output value although three is an output value without any input value corresponding to it it is still a function this could have been written in this form we wrote each of the input values and what output value they correspond to now let's look at this relations this relation is not a function because the input two has more than one output value that's seven and five no input value can have more than one output value this could have been written this way notice we have two written twice with different y values one is seven and the other is five finally let's look at this relations this is a function multiple input values can go to the same output value so here although the input value four and the input value five both gives eight it is a function again this could have been written this way notice we have our y value 8 written twice with this information let's answer answer our question the answer is c c is not a function the input value three has two output values that's six and8 for a function no input value can have more than one output values Happy Thanksgiving get the full course at ultimate algebra.com and change your algebra forever watch more of our videos on YouTube please like share and subscribe