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

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

Identify the expression inside the absolute value: here, \(A = x - 1\) and \(B = 2\).

Set up the two inequalities based on the rule: \(x - 1 \leq -2\) or \(x - 1 \geq 2\).

Solve each inequality separately: For \(x - 1 \leq -2\), add 1 to both sides to get \(x \leq -1\). For \(x - 1 \geq 2\), add 1 to both sides to get \(x \geq 3\).

Combine the solutions to write the final answer as \(x \leq -1\) or \(x \geq 3\).

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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 any 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 expressions without absolute values.

Solving Absolute Value Inequalities

Absolute value inequalities like |x - 1| ≥ 2 split into two cases: either the expression inside is greater than or equal to 2, or less than or equal to -2. This leads to two separate inequalities to solve, reflecting the distance being at least 2 units away from 1 on the number line.

Number Line Interpretation

Interpreting absolute value inequalities on a number line helps visualize the solution set. For |x - 1| ≥ 2, the solutions are all points at least 2 units away from 1, meaning x ≤ -1 or x ≥ 3. This visualization aids in understanding and verifying the solution intervals.