now in this lesson we're going to focus on solving rational equations so let's start with our first example 5 over 8 minus 3 over 5 and let's set that equal to x over 10. what do we need to do in order to find the value of x what would you do the best thing we can do is clear away all fractions we have an eight a five and a ten what is the least common multiple of eight five and ten well we can make a list multiples of five are five ten fifteen and so forth multiples of 8 are 8 16 24 32 and 40. multiples of 10 also include 40. so 40 is the least common multiple let's multiply every fraction by 40. so what's 5 over 8 times 40 you can do 5 times 40 which is 200 and then divide 200 by 8 or you can do 40 divided by 8 which is 5 times the 5 on top and that's going to be 25 now what about three fifths of 40 40 divided by five is eight eight times three is twenty four and the last one forty divided by ten is four times x that's four x twenty five minus twenty four is one and so x is equal to one fourth so that's the answer here's the next problem x plus eight over x is equal to 6. feel free to pause the video and work on this example so what we're going to do in this problem we're going to multiply both sides by x so x times x is x squared and then 8 over x times x the x variables will cancel it's just going to be 8 and 6 times x is 6x now let's move the 6x from the right side to the left side on the right side is positive 6x but on the left side it's going to be negative now we can factor it what two numbers multiply to eight but add up to negative six this is negative four and two so it's x minus four times x minus two and so we can clearly see that x is equal to positive four and positive two and so that's it here's the next one x plus 3 divided by x minus 3 let's say that's equal to 12 over 3. whenever you have two fractions separated by an equal sign what you want to do is you want to cross multiply so 12 times x minus 3 is 12x minus 36 and 3 times x plus 3 that's 3x plus 9. now let's subtract both sides by 3x and let's add 36 to both sides 12x minus 3x is 9x 9 plus 36 is 45 and 45 divided by 9 is 5. so x is equal to 5. try this one nine divided by x is x over four so once again we have two fractions separated by an equal sign let's cross multiply x times x is x squared and 9 times 4 is 36 so all we need to do is take the square root of both sides the square root of 36 is plus or minus 6. so there's two answers positive six and negative six now what about this one four divided by x minus three and let's say that's equal to nine over x plus two so for this problem as well cross multiply so four times x plus two that's four x plus eight and then nine times x minus three that's nine x minus twenty seven so let's subtract both sides by 4x and let's add 28 i mean not 28 but rather 27 to both sides 8 plus 27 that's 35 9 minus 4 is 5. so all we need to do now is divide by 5. 35 divided by 5 is 7. so x is equal to 7. now let's say that we have x plus two divided by three plus four and let's say that's equal to x plus nine divided by two find the value of x the least common multiple of two and three is six so let's multiply everything by six to get rid of the fractions six divided by three is two now let's multiply two by x plus two and that's going to be two x plus four now four times six is twenty four and six divided by two is three and three times x plus nine that's going to be 3x plus 27 so now let's combine 4 and 24 which is 28 now let's subtract both sides by 2x and also by 27 28 minus 27 is one 3x minus 2x is x so therefore x is equal to one here's the next problem four divided by x plus eight divided by x plus two let's set that equal to four find the value of x so in this case the common denominator is x times x plus two if we multiply 4 over x by x x plus 2 the x variables will cancel and so that's going to leave behind 4 times x plus 2 which if we distribute 4 is going to be 4x plus 8. now x plus 2 will cancel leaving behind x times 8 or simply 8x and then here we'll have 4 times x times x plus 2 which is 4x x plus 2. now we can add 4x and 8x that's going to be 12x and now let's distribute the 4x 4x times x is 4x squared 4x times 2 is 8x everything on the left side let's move it to the right side so instead of having positive 12x on the left side it's going to be negative 12x on the right side and 8 is going to change to negative 8. now let's combine like terms 8x minus 12x is negative 4x so now what we need to do is factor we can take out a four and this will leave us with x squared minus x instead of plus x minus two now two numbers that multiply to negative two but at negative one is going to be a negative two and positive one so it's gonna be x minus two times x plus one so if we set each factor equal to zero we can see that x is equal to two and x is equal to negative one and so that's going to be the answer to the problem now let's try the last example 5 or rather i'll take that back not 5 x over x plus 5 minus 5 over x minus 5. let's say that's equal to 14 over x squared minus 25. go ahead and find the value of x now what we should do first is factor x squared minus 25 and that's going to be x plus five times x minus five now we need to clear away all fractions so let's multiply the top well let's multiply both sides the left side and the right side by x plus five times x minus five the common denominator so if we take this fraction and multiply it by these two we can see that x plus five will cancel leaving behind x times x minus five now if we take the second fraction multiply by those two the x minus five term will cancel leaving behind five times x plus five and then x plus 5 will cancel and x minus 5 will cancel leaving 14. so now let's distribute x times x minus 5. that's x squared minus 5x and if we distribute the negative 5 it's going to be negative 5x minus 25. now let's subtract both sides by 14 and let's combine like terms negative 5x and negative 5x that's negative 10x negative 25 minus 14 that's negative 39. what two numbers multiplied to negative 39 but add to negative 10. i'm thinking of negative 13 and 3 so this is going to be x minus 13 x plus 3. so therefore x is equal to 13 and negative 3.