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

Verified step by step guidance

1. Start by breaking the compound inequality into two separate inequalities: $-4\le \frac{x+1}{2}$ and $\frac{x+1}{2}\le 5$.
2. Solve the first inequality $-4\le \frac{x+1}{2}$ by multiplying both sides by 2 to eliminate the fraction: $-8\le x+1$.
3. Subtract 1 from both sides of $-8\le x+1$ to isolate $x$: $-9\le x$.
4. Solve the second inequality $\frac{x+1}{2}\le 5$ by multiplying both sides by 2: $x+1\le 10$.
5. Subtract 1 from both sides of $x+1\le 10$ to isolate $x$: $x\le 9$.

Key Concepts

Inequalities

Inequalities are mathematical statements that express the relationship between two expressions that are not necessarily equal. They use symbols such as '≤' (less than or equal to) and '≥' (greater than or equal to) to indicate the range of values that satisfy the condition. Understanding how to manipulate and solve inequalities is crucial for finding solution sets.

Interval Notation

Interval notation is a way of representing a set of numbers between two endpoints. It uses brackets [ ] to include endpoints and parentheses ( ) to exclude them. For example, the interval [a, b) includes 'a' but not 'b', indicating all numbers from 'a' to 'b', including 'a' and excluding 'b'. This notation is essential for expressing the solution set of inequalities.

Solving Compound Inequalities

A compound inequality consists of two or more inequalities that are combined into one statement, often using 'and' or 'or'. To solve a compound inequality, one must isolate the variable while maintaining the relationships defined by the inequalities. This process often involves breaking the compound inequality into separate parts and solving each part individually before combining the results.