Problem 74 - College Algebra Essentials

Question

Solve each absolute value inequality. $$|5 x-2|>13$$

Step-by-Step Solution

Answer

The solutions to the inequality \(|5x - 2| > 13\) are \(x < -2.2\) or \(x > 3\).

Step 1: Understanding Absolute Value Inequalities

An absolute value inequality is solved by splitting it into two separate inequalities. For an inequality of the form \(|a| > b\), this gives us the two inequalities \(a > b\) and \(a < -b\). In this context, the task is to solve the inequality \(|5x - 2| > 13\), that means, \(5x-2\) will be substituted as \(a\), and \(13\) will be substituted as \(b\).

Step 2: Solving the First Inequality

Solve the inequality where \(5x - 2 > 13\). Add 2 to both sides to isolate \(5x\), giving \(5x > 15\). After that, divide both sides by 5, which yields \(x > 3\).

Step 3: Solving the Second Inequality

Similarly, solve the second inequality where \(5x - 2 < -13\). Add 2 to both sides, resulting in \(5x < -11\). Divide both sides by 5, so the inequality \(x < -2.2\) is obtained.

Step 4: Combining the Solutions

When the two solutions are combined, the complete solution to the inequality \(|5x - 2| > 13\) is \(x < -2.2\) or \(x > 3\). This means that any value of \(x\) that is less than -2.2 or greater than 3 will satisfy the original inequality.