Solve each inequality. Give the solution set in interval notation. | 1/2 - x | ≤ 2

Verified step by step guidance

1. Start by considering the absolute value inequality \(|\frac{1}{2} - x| \leq 2\). This can be split into two separate inequalities: \(\frac{1}{2} - x \leq 2\) and \(\frac{1}{2} - x \geq -2\).
2. Solve the first inequality \(\frac{1}{2} - x \leq 2\) by isolating \(x\). Subtract \(\frac{1}{2}\) from both sides to get \(-x \leq \frac{3}{2}\). Then, multiply both sides by \(-1\) and reverse the inequality sign to get \(x \geq -\frac{3}{2}\).
3. Solve the second inequality \(\frac{1}{2} - x \geq -2\) by isolating \(x\). Subtract \(\frac{1}{2}\) from both sides to get \(-x \geq -\frac{5}{2}\). Then, multiply both sides by \(-1\) and reverse the inequality sign to get \(x \leq \frac{5}{2}\).
4. Combine the solutions from both inequalities. The solution set is \(-\frac{3}{2} \leq x \leq \frac{5}{2}\).
5. Express the solution set in interval notation as \([-\frac{3}{2}, \frac{5}{2}]\).

Key Concepts

Absolute Value Inequality

An absolute value inequality expresses a condition where the distance of a variable from a certain point is constrained. For example, |x - a| ≤ b means that x is within b units of a. To solve such inequalities, we typically break them into two separate inequalities, one for the positive case and one for the negative case.

Interval Notation

Interval notation is a mathematical notation used to represent a range of values. It uses parentheses and brackets to indicate whether endpoints are included or excluded. For instance, (a, b) means all numbers between a and b, excluding a and b, while [a, b] includes both endpoints.

Solving Inequalities

Solving inequalities involves finding the values of a variable that satisfy a given condition. This process often includes isolating the variable, manipulating the inequality, and considering the direction of the inequality sign, especially when multiplying or dividing by negative numbers. The solution is typically expressed in interval notation.