Problem 102 - College Algebra Essentials

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\frac{x^{2}-25}{x-5}=x-5$$

Step-by-Step Solution

Answer

The original statement \(\frac{x^{2}-25}{x-5}=x-5\) is false. The correct statement should be \(\frac{x^{2}-25}{x-5}=x+5\).

Step 1: Factor the Numerator

Start by factoring the numerator of the left hand side of the given equation. The expression \(x^{2}-25\) is a difference of two squares, and it factors to \((x-5)(x+5)\). So, the fraction becomes \(\frac{(x-5)(x+5)}{x-5}\).

Step 2: Simplify the Expression

Simplify the equation by canceling out the common factors in the numerator and the denominator. Here, \(x-5\) is common in both, so the fraction simplifies to \(x+5\).

Step 3: Compare the Simplified Expression with the Original

Now compare the simplified expression on the left side, which is \(x+5\), with the given right side \(x-5\). They aren't the same, so the original statement is false.

Step 4: Correct the Statement

To make the statement true, the right side of the equation should be changed to match the left side. So, the correct statement would be \(\frac{x^{2}-25}{x-5}=x+5\).