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

Verified step by step guidance

1. 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 \(A \geq B\).
2. Identify the expression inside the absolute value: here, \(A = x - 1\) and \(B = 2\).
3. Set up the two inequalities based on the rule: \(x - 1 \leq -2\) or \(x - 1 \geq 2\).
4. 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 \(x \geq 3\).
5. Combine the solutions to write the final answer as \(x \leq -1\) or \(x \geq 3\).

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 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.