Solve each inequality. Give the solution set using interval notation. See Example 10. 10 ≤ 2x + 4 ≤ 16

Verified step by step guidance

1. Start by understanding that the compound inequality \(10 \leq 2x + 4 \leq 16\) means that \$2x + 4$ is simultaneously greater than or equal to 10 and less than or equal to 16.
2. To isolate \(x\), subtract 4 from all parts of the inequality: \(10 - 4 \leq 2x + 4 - 4 \leq 16 - 4\), which simplifies to \(6 \leq 2x \leq 12\).
3. Next, divide all parts of the inequality by 2 to solve for \(x\): \(\frac{6}{2} \leq \frac{2x}{2} \leq \frac{12}{2}\), which simplifies to \(3 \leq x \leq 6\).
4. Interpret the solution: \(x\) is greater than or equal to 3 and less than or equal to 6.
5. Express the solution set in interval notation as \([3, 6]\), which includes both endpoints.

Key Concepts

Compound Inequalities

A compound inequality involves two inequalities joined together, such as 'a ≤ expression ≤ b'. Solving it requires isolating the variable so that the inequality holds true for both parts simultaneously, resulting in a range of values.

Solving Linear Inequalities

Solving linear inequalities involves performing algebraic operations like addition, subtraction, multiplication, or division to isolate the variable. When multiplying or dividing by a negative number, the inequality sign must be reversed.

Interval Notation

Interval notation is a concise way to represent solution sets of inequalities using parentheses and brackets. Brackets [ ] indicate inclusion of endpoints, while parentheses ( ) indicate exclusion, clearly showing the range of valid values.