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

Verified step by step guidance

1. Recognize that the inequality is a compound inequality: \(-9 \leq x + 5 \leq 15\). This means that \(x + 5\) is simultaneously greater than or equal to \(-9\) and less than or equal to \$15$.
2. To isolate \(x\), subtract \$5$ from all three parts of the inequality to maintain the balance. This gives: \(-9 - 5 \leq x + 5 - 5 \leq 15 - 5\).
3. Simplify each part: \(-14 \leq x \leq 10\).
4. Interpret the solution: \(x\) is any number between \(-14\) and \$10$, inclusive of both endpoints because of the 'less than or equal to' signs.
5. Express the solution set in interval notation as \([-14, 10]\).

Key Concepts

Compound Inequalities

Compound inequalities involve two inequalities joined together, often with 'and' or 'or'. In this problem, the inequality -9 ≤ x + 5 ≤ 15 means x + 5 is simultaneously greater than or equal to -9 and less than or equal to 15. Solving requires treating both parts together to find the range of x values that satisfy both conditions.

Solving Linear Inequalities

Solving linear inequalities involves isolating the variable on one side by performing inverse operations, similar to solving equations. When adding, subtracting, multiplying, or dividing, the inequality sign must be preserved or reversed appropriately. Here, subtracting 5 from all parts of the inequality isolates x.

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. After solving the inequality, the solution set is expressed in interval notation to clearly show all values of x that satisfy the inequality.