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Chapter 1: Problem 71

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

Short Answer

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

Step by step solution

01

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.
02

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 \(-x + 1 \geq 2\).
03

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.
04

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

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Most popular questions from this chapter

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