Solve each inequality. Give the solution set using interval notation. 5 ≤ 2x -3 ≤ 7

Verified step by step guidance

1. Start by breaking the compound inequality into two separate inequalities: $5\le 2x-3$ and $2x-3\le 7$.
2. Solve the first inequality $5\le 2x-3$ by adding 3 to both sides to isolate the term with $x$: $5+3\le 2x$.
3. Simplify the inequality to get $8\le 2x$, then divide both sides by 2 to solve for $x$: $4\le x$.
4. Solve the second inequality $2x-3\le 7$ by adding 3 to both sides: $2x\le 7+3$.
5. Simplify the inequality to get $2x\le 10$, then divide both sides by 2 to solve for $x$: $x\le 5$.

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 5 ≤ 2x - 3 ≤ 7 requires solving both parts simultaneously. Understanding how to break down and solve compound inequalities is key to finding the correct solution set.