Solve each inequality. Give the solution set using interval notation. See Example 10. -9 ≤ x + 5 ≤ 15

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Start with the compound inequality: \(-9 \leq x + 5 \leq 15\).

To isolate \(x\), subtract 5 from all three parts of the inequality: \(-9 - 5 \leq x + 5 - 5 \leq 15 - 5\).

Simplify each part: \(-14 \leq x \leq 10\).

Interpret the solution: \(x\) is greater than or equal to \(-14\) and less than or equal to \(10\).

Express the solution set in interval notation as \([-14, 10]\).

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주요 개념

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Compound Inequalities

Compound inequalities involve two inequalities combined into one statement, such as -9 ≤ x + 5 ≤ 15. Solving them requires isolating the variable by performing operations on all parts of the inequality simultaneously to maintain the inequality's truth.

Properties of Inequalities

When solving inequalities, adding, subtracting, multiplying, or dividing all parts by the same positive number preserves the inequality direction. Multiplying or dividing by a negative number reverses the inequality signs. Understanding these properties ensures correct manipulation of inequalities.

Interval Notation

Interval notation expresses solution sets of inequalities concisely using brackets and parentheses. Brackets [ ] indicate inclusion of endpoints, while parentheses ( ) indicate exclusion. For example, [a, b] means all values from a to b, including a and b.