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

Short Answer

Expert verified
The solution to the inequality \(|x-1| \leq 2\) is \(-1 \leq x \leq 3\).

Step by step solution

01

Convert the Absolute Value Inequality into Two Separate Inequalities

First, rewrite the inequality \(|x-1| \leq 2\) as \(-2 \leq x-1 \leq 2\).
02

Solve the Two Inequalities

It is now necessary to solve the two inequalities separately. For \(-2 \leq x-1\), after adding 1 to both sides, the result is \(-1 \leq x\). For \(x-1 \leq 2\), after adding 1 to both sides, the result is \(x \leq 3\).
03

Combine the Results

The solution to the original inequality is the intersection of the two individual solutions. That is, the final result is \(-1 \leq x \leq 3\). This means the solution to the inequality \(|x-1| \leq 2\) consists of all real numbers between -1 and 3, inclusive.

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