Solve each inequality. Give the solution set in interval notation. | 5/3 - (1/2) x | > 2/9

Verified step by step guidance

1. Start by understanding that the inequality involves an absolute value expression: \(\left| \frac{5}{3} - \frac{1}{2}x \right| > \frac{2}{9}\). The absolute value inequality \(|A| > B\) means that either \(A > B\) or \(A < -B\).
2. Set up two separate inequalities based on the definition of absolute value inequalities: 1) \(\frac{5}{3} - \frac{1}{2}x > \frac{2}{9}\) 2) \(\frac{5}{3} - \frac{1}{2}x < -\frac{2}{9}\)
3. Solve the first inequality for \(x\): - Subtract \(\frac{5}{3}\) from both sides. - Multiply both sides by \(-2\) to isolate \(x\) (remember to reverse the inequality sign when multiplying by a negative number).
4. Solve the second inequality for \(x\): - Subtract \(\frac{5}{3}\) from both sides. - Multiply both sides by \(-2\) (again, reverse the inequality sign).
5. Express the solution sets from both inequalities in interval notation and combine them using the union symbol \(\cup\) to represent all \(x\) values that satisfy the original inequality.

Key Concepts

Absolute Value Inequalities

Absolute value inequalities involve expressions where the distance from zero is compared to a number. For |A| > B, the solution splits into two cases: A > B or A < -B. Understanding how to break down and solve these cases is essential for finding the solution set.

Solving Linear Inequalities

Solving linear inequalities requires isolating the variable while maintaining inequality direction. Operations like addition, subtraction, multiplication, or division are applied carefully, especially when multiplying or dividing by negative numbers, which reverses the inequality sign.

Interval Notation

Interval notation expresses solution sets as intervals on the number line, using parentheses for open intervals and brackets for closed intervals. It concisely represents all values satisfying the inequality, including unions when multiple intervals are involved.