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

Verified step by step guidance

1. Start by isolating the inequality by multiplying all parts by 2 to eliminate the fraction: \(-4 \times 2 \leq \frac{x + 1}{2} \times 2 \leq 5 \times 2\).
2. This simplifies to: \(-8 \leq x + 1 \leq 10\).
3. Next, solve the left part of the inequality: \(-8 \leq x + 1\).
4. Subtract 1 from both sides to isolate \(x\): \(-8 - 1 \leq x\).
5. Now, solve the right part of the inequality: \(x + 1 \leq 10\). Subtract 1 from both sides to isolate \(x\): \(x \leq 10 - 1\).

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 ≤ (less than or equal to) and ≥ (greater than or equal to). Solving inequalities involves finding the values of the variable that make the inequality true, which can include manipulating both sides of the inequality while maintaining its direction.

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 in the set. For example, (a, b) means all numbers between a and b, excluding a and b, while [a, b] includes both endpoints. This notation is particularly useful for expressing the solution sets of inequalities.

Fractional Inequalities

Fractional inequalities involve expressions where the variable is in the numerator or denominator of a fraction. To solve these inequalities, it is often necessary to isolate the variable by manipulating the fraction, which may include multiplying or dividing by expressions that could affect the inequality's direction. Understanding how to handle fractions is crucial for accurately solving and interpreting the results.