Solve each inequality. Give the solution set in interval notation. | (2/3)x + 1/2 | ≤ 1/6

Verified step by step guidance

1. Start by isolating the absolute value expression: \( \left| \frac{2}{3}x + \frac{1}{2} \right| \leq \frac{1}{6} \).
2. This inequality can be split into two separate inequalities: \( \frac{2}{3}x + \frac{1}{2} \leq \frac{1}{6} \) and \( \frac{2}{3}x + \frac{1}{2} \geq -\frac{1}{6} \).
3. Solve the first inequality: \( \frac{2}{3}x + \frac{1}{2} \leq \frac{1}{6} \). Subtract \( \frac{1}{2} \) from both sides to isolate the term with \( x \).
4. Solve the second inequality: \( \frac{2}{3}x + \frac{1}{2} \geq -\frac{1}{6} \). Again, subtract \( \frac{1}{2} \) from both sides to isolate the term with \( x \).
5. Combine the solutions from both inequalities to find the solution set in interval notation.

Key Concepts

Absolute Value Inequalities

Absolute value inequalities involve expressions that measure the distance of a number from zero on the number line. The inequality |A| ≤ B means that A lies within the interval [-B, B]. Understanding how to manipulate these inequalities is crucial for solving them correctly.

Solving Linear Inequalities

Solving linear inequalities involves isolating the variable on one side of the inequality sign. This process is similar to solving linear equations but requires special attention to the direction of the inequality sign, especially when multiplying or dividing by negative numbers.

Interval Notation

Interval notation is a way of representing a set of numbers between two endpoints. It uses parentheses and brackets to indicate whether endpoints are included (closed interval) or excluded (open interval). Understanding how to express solution sets in this format is essential for conveying the results of inequalities.