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

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Recall that an absolute value inequality of the form \(|A| > B\) (where \(B > 0\)) can be rewritten as two separate inequalities: \(A > B\) or \(A < -B\).

Identify the expression inside the absolute value: here, \(A = 3x - 8\) and \(B = 7\).

Set up the two inequalities based on the rule: \(3x - 8 > 7\) or \(3x - 8 < -7\).

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 \(x < \frac{1}{3}\).

Combine the solutions to express the final answer: \(x < \frac{1}{3}\) or \(x > 5\).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.