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

검증된 단계별 안내

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.

To isolate \(x\), subtract 4 from all parts of the inequality: \(10 - 4 \leq 2x + 4 - 4 \leq 16 - 4\), which simplifies to \(6 \leq 2x \leq 12\).

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 \(3 \leq x \leq 6\).

Interpret the solution: \(x\) is greater than or equal to 3 and less than or equal to 6.

Express the solution set in interval notation as \([3, 6]\), which includes both endpoints.

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주요 개념

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

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.