Solve each inequality. Give the solution set in interval notation. 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 \(x\), first subtract 4 from all three 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 \(3 \leq x \leq 6\) as all real numbers \(x\) between 3 and 6, including the endpoints 3 and 6.

Express the solution set in interval notation as \([3, 6]\), where the square brackets indicate that the endpoints are included.

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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 10 ≤ 2x + 4 ≤ 16. Solving it requires finding all values of the variable that satisfy both inequalities simultaneously, often by isolating the variable within the combined inequality.

Solving Linear Inequalities

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

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. For example, [3, 6) means all numbers from 3 to 6, including 3 but excluding 6.