Problem 60 - Introductory Algebra for College Students

Question

Simplify each rational expression. If the rational expression cannot be simplified, so state. $$\frac{x-7}{7-x}$$

Step-by-Step Solution

Answer

The simplified form of the given rational expression \(\frac{x-7}{7-x}\) is -1.

Step 1: Observing the expression and seeing structure

Looking at the given fraction, \(\frac{x-7}{7-x}\), the numerator is \(x-7\) and the denominator is \(7-x\). Both expressions are the same but the order of the terms is different.

Step 2: Simplifying

Since \(x-7\) and \(7-x\) are the same expressions but in different order, multiplying the denominator \(7-x\) by -1 gives \(-(7-x) = x-7\). So we can write the given expression as \(\frac{x-7}{-(x-7)}\). Now, since \(x-7\) divided by \(x-7\) is 1, the given expression simplifies to \(\frac{1}{-1}\).

Step 3: Final Simplification

\(\frac{1}{-1}\) simplifies to -1.