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

Verified step by step guidance

1. Understand that the absolute value inequality \(|x - 1| \leq 2\) means the expression inside the absolute value, \(x - 1\), is within 2 units of 0 on the number line.
2. Rewrite the absolute value inequality \(|x - 1| \leq 2\) as a compound inequality: \(-2 \leq x - 1 \leq 2\).
3. Solve the compound inequality by first adding 1 to all parts of the inequality: \(-2 + 1 \leq x - 1 + 1 \leq 2 + 1\).
4. Simplify the inequality to get \(-1 \leq x \leq 3\).
5. Interpret the solution: The values of \(x\) that satisfy the inequality are all the numbers between \(-1\) and \(3\), inclusive.

Key Concepts

Absolute Value

Absolute value measures the distance of a number from zero on the number line, regardless of direction. For any real number x, the absolute value is denoted as |x| and is defined as |x| = x if x ≥ 0, and |x| = -x if x < 0. Understanding absolute value is crucial for solving inequalities that involve it, as it transforms the problem into a range of values.

Inequalities

Inequalities express a relationship between two expressions that are not necessarily equal. They can be strict (using < or >) or non-strict (using ≤ or ≥). In the context of absolute value inequalities, the goal is to find the set of values that satisfy the inequality, which often involves breaking it down into two separate cases based on the definition of absolute value.

Solving Absolute Value Inequalities

To solve an absolute value inequality like |x - 1| ≤ 2, one must consider the two scenarios that arise from the definition of absolute value. This leads to two separate inequalities: x - 1 ≤ 2 and -(x - 1) ≤ 2. Solving these inequalities will yield a range of solutions, which can be expressed in interval notation to represent all possible values of x that satisfy the original inequality.