Solve each inequality. Give the solution set using interval notation. See Example 10.
- 4 ≤ (x + 1)/2 ≤ 5

Verified step by step guidance

1. Start by writing the compound inequality clearly: \(-4 \leq \frac{x + 1}{2} \leq 5\).
2. To eliminate the fraction, multiply all three parts of the inequality by 2 (which is positive, so the inequality signs remain the same): \(-4 \times 2 \leq x + 1 \leq 5 \times 2\).
3. Simplify the multiplication: \(-8 \leq x + 1 \leq 10\).
4. Next, isolate \(x\) by subtracting 1 from all parts of the inequality: \(-8 - 1 \leq x + 1 - 1 \leq 10 - 1\).
5. Simplify the expressions to get the solution for \(x\): \(-9 \leq x \leq 9\). Express this solution in interval notation as \([-9, 9]\).

Key Concepts

Solving Compound Inequalities

A compound inequality involves two inequalities joined by 'and' or 'or'. To solve, treat it as two separate inequalities and find the values of the variable that satisfy both simultaneously. The solution is the intersection of the solution sets for each inequality.

Manipulating Inequalities with Fractions

When solving inequalities involving fractions, multiply or divide both sides by the denominator carefully, ensuring it is positive to avoid reversing the inequality sign. Simplify the expression step-by-step to isolate the variable while maintaining inequality direction.

Interval Notation

Interval notation expresses the solution set of inequalities using parentheses and brackets to denote open or closed intervals. Parentheses indicate values not included, while brackets include endpoints. It provides a concise way to represent all values satisfying the inequality.