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Chapter 7: Problem 59

Simplify each rational expression. If the rational expression cannot be simplified, so state. $$\frac{x-5}{5-x}$$

Short Answer

Expert verified
The simplified version of the rational expression \(\frac{x-5}{5-x}\) is -1.

Step by step solution

01

Identify the rational expression

The first step is to identify the rational expression given, which is \(\frac{x-5}{5-x}\). Rational expressions are simply fractions in which the numerator and/or the denominator are polynomials.
02

Use properties of negatives to simplify the expression

Since the numerator and denominator are opposites, meaning they contain the same terms but with opposite signs, we know that \((x-5) = -(5-x)\). Using this property, we can express the denominator as -1 times the numerator: \(-1(x-5)\).
03

Simplify the expression

Now, we have \(\frac{x-5}{-1(x-5)}\). We can now see that the numerator and denominator are the same, aside from the negative sign in the denominator. When we divide a number by itself, we get 1, so we are left with \(\frac{1}{-1}\), which simplifies to -1.
04

Verify the final solution

Finally, ensure that the simplified expression is correct. In this case, the simplified expression is -1, which is a vastly less complex version of the original rational expression.

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