Problem 59 - Introductory Algebra for College Students

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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.

Step 2: 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: \(-1(x-5)\).

Step 3: 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.

Step 4: 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.