Problem 78 - College Algebra Essentials

Question

Absolute value expressions are equal when the expressions inside the absolute value bars are equal to or opposites of each other. $$ |2 x-7|=|x+3| $$

Step-by-Step Solution

Answer

Only \(x = 10\) is a solution to the equation.

Step 1: Set Up Two Equations

The first step is to set up two equivalent equations, each representing one case of the absolute values. This will give two equations: 1) When both expressions are equal: \(2x - 7 = x + 3\)2) When the expressions are opposites: \(2x - 7 = - (x + 3)\)

Step 2: Simplify Both Equations

Next, solve each equation separately. 1) Subtract x from both sides of the first equation to isolate x on the left side to get \(x = 10\)2) In the second equation, distribute the negative sign to get \(2x - 7 = -x - 3\). Adding x to both sides and 3 to both sides to isolate x will give \(3x = 4\), so \(x = \frac{4}{3}\)

Step 3: Verify The Solutions

Verify both solutions in the original equation. Substituting \(x = 10\) in the original absolute value equation, both sides yield 13, so \(x = 10\) is a solution. However, substituting \(x = \frac{4}{3}\), the left side yields \(|2(\frac{4}{3}) - 7| = |-\frac{1}{3}|\) which does not equal to the right side \(|\frac{4}{3} + 3| = |\frac{13}{3}|\). So \(x = \frac{4}{3}\) is not a solution.