In Exercises 59–94, solve each absolute value inequality. 3|x - 1| + 2 ≥ 8

Verified step by step guidance

1. Start by isolating the absolute value expression. Subtract 2 from both sides of the inequality: \(3|x - 1| + 2 - 2 \geq 8 - 2\).
2. Simplify the inequality: \(3|x - 1| \geq 6\).
3. Divide both sides by 3 to further isolate the absolute value: \(|x - 1| \geq 2\).
4. Recall that an absolute value inequality of the form \(|A| \geq B\) can be split into two separate inequalities: \(A \geq B\) or \(A \leq -B\).
5. Apply this to the inequality \(|x - 1| \geq 2\), resulting in two inequalities: \(x - 1 \geq 2\) or \(x - 1 \leq -2\).

Key Concepts

Absolute Value

Absolute value represents 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|, which equals x if x is non-negative and -x if x is negative. Understanding absolute value is crucial for solving inequalities that involve expressions within absolute value symbols.

Inequalities

Inequalities express a relationship between two expressions that are not necessarily equal. They can be represented using symbols such as >, <, ≥, or ≤. Solving inequalities often involves finding the range of values that satisfy the condition, which may require considering multiple cases, especially when absolute values are involved.

Solving Absolute Value Inequalities

To solve an absolute value inequality, one must isolate the absolute value expression and then set up two separate inequalities based on the definition of absolute value. For example, if |A| ≥ B, it leads to A ≥ B or A ≤ -B. This process allows for determining the solution set that satisfies the original inequality.