In Exercises 59–94, solve each absolute value inequality. 4 + |3 - x/3| ≥ 9

Verified step by step guidance

1. Start by isolating the absolute value expression. Subtract 4 from both sides of the inequality: \(|3 - \frac{x}{3}| \geq 5\).
2. Consider the definition of absolute value: \(|A| \geq B\) implies \(A \geq B\) or \(A \leq -B\). Apply this to the inequality: \(3 - \frac{x}{3} \geq 5\) or \(3 - \frac{x}{3} \leq -5\).
3. Solve the first inequality: \(3 - \frac{x}{3} \geq 5\). Subtract 3 from both sides: \(-\frac{x}{3} \geq 2\). Then multiply both sides by -3, remembering to reverse the inequality sign: \(x \leq -6\).
4. Solve the second inequality: \(3 - \frac{x}{3} \leq -5\). Subtract 3 from both sides: \(-\frac{x}{3} \leq -8\). Then multiply both sides by -3, remembering to reverse the inequality sign: \(x \geq 24\).
5. Combine the solutions from both inequalities. The solution set is \(x \leq -6\) or \(x \geq 24\).

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 'a', the absolute value is denoted as |a| and is defined as |a| = a if a ≥ 0, and |a| = -a if a < 0. Understanding absolute value is crucial for solving inequalities that involve expressions within absolute value bars.

Inequalities

Inequalities express a relationship between two expressions that are not necessarily equal. They can be strict (using < or >) or non-strict (using ≤ or ≥). When solving inequalities, especially those involving absolute values, it is important to consider the different cases that arise from the definition of absolute value, leading to multiple potential solutions.

Solving Absolute Value Inequalities

To solve an absolute value inequality, one must isolate the absolute value expression and then break it into two separate inequalities based on its definition. For example, if |A| ≥ B, it leads to two cases: A ≥ B or A ≤ -B. This process allows for finding all possible solutions that satisfy the original inequality, which may include intervals on the number line.