In Exercises 59–94, solve each absolute value inequality. 3 ≤ |2x - 1|

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Recall that the absolute value inequality \(3 \leq |2x - 1|\) means the expression inside the absolute value, \(2x - 1\), is at least 3 units away from 0 on the number line.

Rewrite the inequality \(3 \leq |2x - 1|\) as two separate inequalities to remove the absolute value: \(2x - 1 \leq -3\) or \(2x - 1 \geq 3\).

Solve the first inequality \(2x - 1 \leq -3\) by adding 1 to both sides: \(2x \leq -2\), then divide both sides by 2 to get \(x \leq -1\).

Solve the second inequality \(2x - 1 \geq 3\) by adding 1 to both sides: \(2x \geq 4\), then divide both sides by 2 to get \(x \geq 2\).

Combine the two solution sets to express the final answer: \(x \leq -1\) or \(x \geq 2\).

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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 ≥ 0, and -A if A < 0. Understanding this helps in rewriting absolute value inequalities into equivalent compound inequalities.

Solving Absolute Value Inequalities

To solve inequalities involving absolute values, such as |A| ≥ k, where k ≥ 0, split the inequality into two cases: A ≥ k or A ≤ -k. This approach transforms the absolute value inequality into two linear inequalities that can be solved separately.

Compound Inequalities and Solution Sets

After splitting the absolute value inequality, the solution is the union of the solution sets of the two inequalities. Understanding how to combine these sets correctly is essential to express the final answer, often in interval notation, representing all values satisfying the original inequality.