In Exercises 59–94, solve each absolute value inequality.|3 - (2/3)x| > 5

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1. Start by understanding the absolute value inequality \(|3 - \frac{2}{3}x| > 5\). This means the expression inside the absolute value, \(3 - \frac{2}{3}x\), is either greater than 5 or less than -5.
2. Set up the two separate inequalities: \(3 - \frac{2}{3}x > 5\) and \(3 - \frac{2}{3}x < -5\).
3. Solve the first inequality: \(3 - \frac{2}{3}x > 5\). Subtract 3 from both sides to get \(-\frac{2}{3}x > 2\). Then, multiply both sides by \(-\frac{3}{2}\) and remember to reverse the inequality sign.
4. Solve the second inequality: \(3 - \frac{2}{3}x < -5\). Subtract 3 from both sides to get \(-\frac{2}{3}x < -8\). Then, multiply both sides by \(-\frac{3}{2}\) and remember to reverse the inequality sign.
5. Combine the solutions from both inequalities to find the solution set for \(|3 - \frac{2}{3}x| > 5\).

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 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 absolute value inequalities, it is important to consider the two cases that arise from the definition of absolute value, leading to two separate inequalities to solve.

Solving Linear Inequalities

Solving linear inequalities involves finding the values of the variable that satisfy the inequality. This process often includes isolating the variable on one side of the inequality sign. When dealing with absolute value inequalities, after breaking them into two cases, one must solve each resulting linear inequality and then combine the solutions to find the overall solution set.