In Exercises 59–94, solve each absolute value inequality. |2x - 6| < 8

Verified step by step guidance

1. Recognize that the inequality involves an absolute value expression: \(|2x - 6| < 8\). The absolute value inequality \(|A| < B\) means that the expression inside the absolute value, \(A\), lies between \(-B\) and \(B\).
2. Rewrite the inequality without the absolute value as a compound inequality: \(-8 < 2x - 6 < 8\).
3. Solve the compound inequality step-by-step. First, add 6 to all three parts: \(-8 + 6 < 2x - 6 + 6 < 8 + 6\), which simplifies to \(-2 < 2x < 14\).
4. Next, divide all parts of the inequality by 2 to isolate \(x\): \(\frac{-2}{2} < \frac{2x}{2} < \frac{14}{2}\), which simplifies to \(-1 < x < 7\).
5. Interpret the solution: \(x\) must be greater than \(-1\) and less than \(7\) to satisfy the original absolute value inequality.

Key Concepts

Absolute Value Definition

The absolute value of a number represents its distance from zero on the number line, always as a non-negative value. For any expression |A| < B, where B > 0, it means the expression A lies between -B and B.

Solving Absolute Value Inequalities

To solve inequalities like |A| < B, rewrite them as a compound inequality: -B < A < B. This allows you to solve for the variable by isolating it within these bounds, providing the solution set.

Linear Inequalities

Linear inequalities involve expressions where variables are to the first power. After removing the absolute value, solving the resulting linear inequalities requires isolating the variable and understanding inequality properties.