In Exercises 59–94, solve each absolute value inequality. |3x - 8| > 7

Verified step by step guidance

1. Start by understanding the absolute value inequality: |3x - 8| > 7. This means the expression inside the absolute value, 3x - 8, is either greater than 7 or less than -7.
2. Set up the two separate inequalities: 3x - 8 > 7 and 3x - 8 < -7.
3. Solve the first inequality: 3x - 8 > 7. Add 8 to both sides to isolate the term with x, resulting in 3x > 15.
4. Divide both sides of the inequality 3x > 15 by 3 to solve for x, giving x > 5.
5. Solve the second inequality: 3x - 8 < -7. Add 8 to both sides to isolate the term with x, resulting in 3x < 1. Then, divide both sides by 3 to solve for x, giving x < \frac{1}{3}.

Key Concepts

Absolute Value

Absolute value represents the distance of a number from zero on the number line, regardless of direction. For any real number 'a', the absolute value is denoted as |a| and is defined as |a| = a if a ≥ 0, and |a| = -a if a < 0. Understanding absolute value is crucial for solving inequalities that involve expressions within absolute value bars.

Inequalities

Inequalities express a relationship between two expressions that are not necessarily equal. They can be strict (using < or >) or non-strict (using ≤ or ≥). When solving absolute value inequalities, it is important to recognize that they can lead to two separate cases, reflecting the definition of absolute value, which can result in two distinct inequalities to solve.

Solving Absolute Value Inequalities

To solve an absolute value inequality like |3x - 8| > 7, one must break it into two separate inequalities: 3x - 8 > 7 and 3x - 8 < -7. Each inequality is then solved independently to find the solution set. This process highlights the concept that the expression inside the absolute value can either exceed a positive threshold or fall below a negative threshold.