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

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Recognize that the inequality involves an absolute value: \(|\frac{1}{2} - x| \leq 2\). Recall that for any expression \(A\), \(|A| \leq B\) means \(-B \leq A \leq B\).

Apply this property to the inequality: \(-2 \leq \frac{1}{2} - x \leq 2\).

Solve the compound inequality by isolating \(x\). Start with the left part: \(-2 \leq \frac{1}{2} - x\). Subtract \(\frac{1}{2}\) from both sides to get \(-2 - \frac{1}{2} \leq -x\).

Multiply both sides of the inequality by \(-1\) to solve for \(x\). Remember to reverse the inequality signs when multiplying by a negative number. This gives \(x \leq 2 + \frac{1}{2}\).

Now solve the right part of the compound inequality: \(\frac{1}{2} - x \leq 2\). Subtract \(\frac{1}{2}\) from both sides to get \(-x \leq 2 - \frac{1}{2}\). Multiply both sides by \(-1\) (and reverse the inequality) to get \(x \geq -2 + \frac{1}{2}\). Combine both results to write the solution set in interval notation.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Absolute Value Inequalities

Absolute value inequalities involve expressions where the distance of a number from zero is compared to another value. For inequalities like |A| ≤ B, the solution includes all values of A between -B and B, inclusive. Understanding how to rewrite and solve these inequalities is essential.

Solving Linear Inequalities

Solving linear inequalities requires isolating the variable on one side while maintaining the inequality's direction. When dealing with compound inequalities, solutions are often expressed as intervals. Mastery of inequality properties helps in correctly manipulating and solving these expressions.

Interval Notation

Interval notation is a concise way to represent sets of numbers between two endpoints. Square brackets [ ] indicate inclusion of endpoints, while parentheses ( ) indicate exclusion. Using interval notation correctly communicates the solution set of inequalities clearly and efficiently.