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

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1. Recall that an absolute value inequality of the form \(|A| \leq B\) means that the expression inside the absolute value, \(A\), lies between \(-B\) and \(B\). So, rewrite the inequality \(|x - 1| \leq 2\) as a compound inequality: \(-2 \leq x - 1 \leq 2\).
2. Next, solve the compound inequality by isolating \(x\) in the middle. Add 1 to all three parts of the inequality: \(-2 + 1 \leq x - 1 + 1 \leq 2 + 1\), which simplifies to \(-1 \leq x \leq 3\).
3. Interpret the solution: \(x\) can be any value between \(-1\) and \(3\), inclusive, because the original inequality uses the 'less than or equal to' symbol.
4. Express the solution in interval notation as \([-1, 3]\), which represents all real numbers \(x\) such that \(-1 \leq x \leq 3\).
5. Optionally, you can graph the solution on a number line by shading the region between \(-1\) and \(3\), including the endpoints.

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 real number x, |x| equals x if x is positive or zero, and -x if x is negative. Understanding this helps interpret expressions like |x - 1| as the distance between x and 1.

Solving Absolute Value Inequalities

An inequality involving absolute value, such as |A| ≤ B, can be rewritten as a compound inequality: -B ≤ A ≤ B, provided B is non-negative. This allows the absolute value inequality to be solved by considering the range of values A can take within these bounds.

Compound Inequalities and Interval Notation

After rewriting the absolute value inequality as a compound inequality, solving it involves isolating the variable and expressing the solution as an interval. Interval notation concisely represents all values satisfying the inequality, for example, x ∈ [a, b], indicating x lies between a and b inclusive.