Solve each inequality. Give the solution set in interval notation. 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 \(x\), first subtract 4 from all three parts of the inequality: \(10 - 4 \leq 2x + 4 - 4 \leq 16 - 4\), which simplifies to \(6 \leq 2x \leq 12\).
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 \(3 \leq x \leq 6\) as all real numbers \(x\) between 3 and 6, including the endpoints 3 and 6.
5. Express the solution set in interval notation as \([3, 6]\), where the square brackets indicate that the endpoints are included.

Key Concepts

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.