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

Verified step by step guidance

1. Recall that an absolute value inequality of the form \(|A| > B\) (where \(B > 0\)) can be rewritten as two separate inequalities: \(A > B\) or \(A < -B\).
2. Identify the expression inside the absolute value: here, \(A = 3x - 8\) and \(B = 7\).
3. Set up the two inequalities based on the rule: \$3x - 8 > 7\( or \)3x - 8 < -7$.
4. Solve each inequality separately: For \$3x - 8 > 7\(, add 8 to both sides to get \)3x > 15\(, then divide both sides by 3 to find \)x > 5\(. For \)3x - 8 < -7\(, add 8 to both sides to get \)3x < 1\(, then divide both sides by 3 to find \)x < \frac{1}{3}$.
5. Combine the solutions to express the final answer: \(x < \frac{1}{3}\) or \(x > 5\).

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 an expression |A|, it equals A if A is non-negative, and -A if A is negative. Understanding this helps in rewriting absolute value inequalities into equivalent compound inequalities.

Solving Absolute Value Inequalities

An inequality involving absolute value, such as |A| > B, can be split into two separate inequalities: A > B or A < -B, when B is positive. This approach allows solving for the variable by considering both cases where the expression inside the absolute value is greater than B or less than -B.

Compound Inequalities and Solution Sets

After splitting the absolute value inequality, the solutions form a compound inequality, often expressed as a union of intervals. Understanding how to combine these intervals and represent the solution set on a number line is essential for interpreting and communicating the final answer.