Problem 61 - College Algebra Essentials

Question

Solve each absolute value inequality. $$|x-1| \leq 2$$

Step-by-Step Solution

Answer

The solution for the inequality is \(-1 \leq x \leq 3\).

Step 1: Case 1: Nonnegative \(x-1\)

This can be solved by isolating \(x\) in the inequality \(x-1 \leq 2\). This gives: \(x \leq 2+1\), thus the solution is \(x \leq 3\).

Step 2: Case 2: Negative \(x-1\)

Likewise, this can be solved by isolating \(x\) in the inequality \(x-1 \geq -2\). This gives: \(x \geq -2+1\), so the solution is \(x \geq -1\).

Step 3: Combine the solutions

Since both conditions must be satisfied at the same time, combine the solution ranges. \(x\) must be less than or equal to 3 and greater than or equal to -1. Therefore, the solution is \(-1 \leq x \leq 3\).