Problem 63 - College Algebra Essentials

Question

Solve each absolute value inequality. $$|2 x-6|<8$$

Step-by-Step Solution

Answer

\(-1 < x < 7\)

Step 1: Setting Up the Inequality Without the Absolute Value

First, the inequality without the absolute value symbols can be written two ways: \(2x - 6 < 8\) (for when the expression inside the absolute value is positive) and \(2x - 6 > -8\) (for when the expression inside the absolute value is negative). These two inequalities come directly from the definition that -b < a < b when |a| < b.

Step 2: Solving the Inequalities

Next, solve each inequality. Add 6 to both sides of the inequality \(2x - 6 < 8\) to obtain \(2x < 14\). After that, divide each side by 2 to find \(x < 7\). For the inequality \(2x - 6 > -8\), add 6 to both sides to get \(2x > -2\), then divide each side by 2 to obtain \(x > -1\). These are the solutions to the original inequality.

Step 3: Writing the Final Solution

The final solution is found by combining the two inequalities. Thus, the solution to the original inequality \(|2x - 6| < 8\) is \(-1 < x < 7\). This comprises all x-values that satisfy both inequalities arrived at in Step 2.