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

Verified step by step guidance

1. Start by understanding that the compound inequality \(5 \leq 2x - 3 \leq 7\) means that \$2x - 3$ is simultaneously greater than or equal to 5 and less than or equal to 7.
2. To isolate \(x\), first add 3 to all three parts of the inequality to eliminate the \(-3\): \(5 + 3 \leq 2x - 3 + 3 \leq 7 + 3\), which simplifies to \(8 \leq 2x \leq 10\).
3. Next, divide all parts of the inequality by 2 to solve for \(x\): \(\frac{8}{2} \leq \frac{2x}{2} \leq \frac{10}{2}\), which simplifies to \(4 \leq x \leq 5\).
4. Interpret the solution: \(x\) is greater than or equal to 4 and less than or equal to 5.
5. Express the solution set in interval notation as \([4, 5]\), which includes both endpoints since the inequalities are inclusive.

Key Concepts

Compound Inequalities

A compound inequality involves two inequalities joined together, often with 'and' or 'or'. In this problem, the inequality 5 ≤ 2x - 3 ≤ 7 means both 5 ≤ 2x - 3 and 2x - 3 ≤ 7 must be true simultaneously. Solving compound inequalities requires handling both parts to find the values of x that satisfy both conditions.

Solving Linear Inequalities

Solving linear inequalities involves isolating the variable on one side by performing inverse operations, similar to solving equations. When multiplying or dividing by a negative number, the inequality sign reverses. The goal is to find all values of x that make the inequality true.

Interval Notation

Interval notation is a concise way to represent solution sets of inequalities using parentheses and brackets. Brackets [ ] indicate that an endpoint is included (closed interval), while parentheses ( ) mean it is excluded (open interval). For example, [a, b] includes all numbers from a to b, including a and b.