Question 684339: (3)/(x+1)- (12x)/3(x+1)+(3)/2(x+1)
Answer by RedemptiveMath(80)   (Show Source):
You can put this solution on YOUR website! We can start this problem off by looking at the denominators. Since they are not equal, we must make them equal by manipulating them. These rational problems can be manipulated just like normal fractions that have unequal denominators. To find the sum of 2/3 and 1/2, we must get a common denominator. 6 is the least common denominator (LCD), so we'll use that. 2/3 + 1/2 = 4/6 + 3/6 = 7/6 or 1 1/6.
Do you remember how we can change 2/3 and 1/2 into 4/6 and 3/6 respectively? We must multiply the top and bottom by the same number to get the desired fraction. However, the number we multiply by is not the same for each fraction. There are more technical ways of writing this out, but they are irrelevant if you already know the way of getting common denominators.
Now, we must put this manipulation into work for rational problems. Having the denominators (x+1), 3(x+1), and 2(x+1), we can see the common piece (x+1). However, the second denominator has 3 of (x+1) and the third has 2. The first only has 1. So, we must multiply the denominators so that the number outside of the parentheses is the same for each fraction. Looking at those numbers, we can see that the common multiple of 1, 3, and 2 is 6. To get to 6 for the first fraction, we must multiply the top and bottom by 6 (1 * 6 = 6). The second fraction we must multiply the top and bottom by 2, and the third by 3. Remember that the numerator and denominator must both be multiplied by the number. Here's the work:
(3)/(x+1) - (12x)/[3(x+1)] + (3)/[2(x+1)]
(6*3)/[6*1(x+1)] - (2*12x)/[2*3(x+1)] + (3*3)/[3*2(x+1)]
18/[6(x+1)] - 24x/[6(x+1)] + 9/[6(x+1)].
Now that we have the common denominator 6(x+1), we can add or subtract the numerators accordingly. Note that it may be best to leave 6(x+1) factored and not use the distributive property yet. We may not even need it.
(18-24x+9)/[6(x+1)] = (27-24x)/[6(x+1)] = [3(9-8x)]/[6(x+1)] = (-8x+9)/[2(x+1)]
OR
(27-24x)/[6(x+1)] = [-3(8x-9)]/[6(x+1)] = -(8x-9)/[2(x+1)].
Depending on your teacher, one of these may be taken as incorrect. Conventions in algebra would like for you to factor negatives out if they are on the coefficient of the variable, but this is your teacher's call. Notice that we can reduce 3 and 6 because they are on the outside of the parentheses. You could also have reduced (-12x)/[3(x+1)] to (-4x)/(x+1) right off the bat. This can be done because the 3 is outside the parentheses and it is not being added or subtracted. Furthermore, 12x is not being added or subtracted. These two terms are only in the processes of division and multiplication. Since they are under simple multiplication or division, we could have reduced them. Then we could have divided the third fraction by 2 to get 4.5/(x+1). We wouldn't have had to worry about the LCD in this case. We would have gotten (4.5-4x)/(x+1), but convention would like there to be no decimals in fractions. So, multiplying by 2 to change 4.5 to 9, we would have had (9-8x)/[2(x+1)] if we did it this way. Nonetheless, we still arrived at this simplified expression the other way.
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