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

Verified step by step guidance

1. Start by breaking the compound inequality into two separate inequalities: \$10 \leq 2x + 4\( and \)2x + 4 \leq 16$.
2. Solve the first inequality \$10 \leq 2x + 4\( by subtracting 4 from both sides to get \)6 \leq 2x$.
3. Divide both sides of the inequality \$6 \leq 2x\( by 2 to isolate \)x\(, resulting in \)3 \leq x$.
4. Solve the second inequality \$2x + 4 \leq 16\( by subtracting 4 from both sides to get \)2x \leq 12$.
5. Divide both sides of the inequality \$2x \leq 12\( by 2 to isolate \)x\(, resulting in \)x \leq 6\(. Combine the solutions from both inequalities to express the solution set in interval notation: \)[3, 6]$.

Key Concepts

Inequalities

Inequalities are mathematical statements that express the relationship between two expressions that are not necessarily equal. They can be represented using symbols such as <, >, ≤, and ≥. Understanding how to manipulate inequalities is crucial for solving them, as the rules differ slightly from those for equations, especially when multiplying or dividing by negative numbers.

Interval Notation

Interval notation is a way of representing a set of numbers between two endpoints. It uses parentheses and brackets to indicate whether the endpoints are included (closed interval) or excluded (open interval). For example, the interval [a, b] includes both a and b, while (a, b) does not. This notation is essential for expressing the solution set of inequalities succinctly.

Compound Inequalities

Compound inequalities involve two or more inequalities that are combined into one statement, often using the conjunction 'and' or 'or'. In the given question, the compound inequality 10 ≤ 2x + 4 ≤ 16 requires solving for x in both parts simultaneously. Understanding how to isolate the variable in compound inequalities is key to finding the correct solution set.