Problem 71 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

The solution to this inequality is \(x \leq -1 \) or \(x \geq 3\).

Step 1: Identify the Expression Inside the Absolute Value

The expression inside the absolute value marks is \(x - 1\). This expression can be either positive or negative and still satisfy the original absolute value inequality.

Step 2: Create Two Separate Inequalities

Create two separate inequalities from the original inequality \(|x-1| \geq 2\). These two inequalities are \(x - 1 \geq 2\) and \(-1 \times (x-1) \geq 2 \), which simplifies to \(-x + 1 \geq 2\).

Step 3: Solve Each Inequality

For the first inequality \(x - 1 \geq 2\), add 1 to both sides to isolate \(x\), obtaining \(x \geq 3\). For the second inequality \(-x + 1 \geq 2\), subtract 1 from both sides and then multiply by -1 to yield \(x \leq -1\). Flip the inequality sign whenever multiplying or dividing by a negative number.

Step 4: Combine the Results

The solutions represent \(x\) values that are greater or equal to 3 and lesser or equal to -1. This is represented by the combined inequality \(x \leq -1 \) or \(x \geq 3\), which encompasses all the valid solutions