Question

You can use \(x-3\) as the LCD when finding the sum \(\frac{5}{x-3}+\frac{2}{3-x}\) What number can you multiply the numerator and the denominator of the second fraction by to get an equivalent fraction with \(x-3\) as the new denominator?

Short answer

-1

Step-by-step solution

Transformation of the denominator

A first step in solving this problem is transforming the \(-3+x\) term (that makes it the additive inverse of \(x-3\)) to become \(x-3\) without changing the mathematical essence. You should focus on how the sign changes to positive when we multiply with a negative number.

Find the number that converts the denominator to \(x - 3\)

It can be noticed that multiplying \(-3+x\) or \(3-x\) by \(-1\), the result will be \(x-3\). Thus, \(-1\) is the number that makes the denominator \(x-3\).

Apply the same operation to the numerator

To keep the fraction equivalent, the same operation (multiplication by -1) must be applied to the numerator of the fraction. Therefore, the new numerator will be \(-2\).