Solve each rational inequality. Give the solution set in interval notation. See Examples 8 and 9. 5/(x+1)>12/(x+1)

Verified step by step guidance

1. Identify the inequality: \( \frac{5}{x+1} > \frac{12}{x+1} \).
2. Subtract \( \frac{12}{x+1} \) from both sides to get: \( \frac{5}{x+1} - \frac{12}{x+1} > 0 \).
3. Combine the fractions: \( \frac{5 - 12}{x+1} > 0 \), which simplifies to \( \frac{-7}{x+1} > 0 \).
4. Determine when the fraction \( \frac{-7}{x+1} \) is positive. Since \(-7\) is negative, the fraction is positive when \(x+1\) is negative.
5. Solve the inequality \(x+1 < 0\) to find the solution set.

Key Concepts

Rational Inequalities

Rational inequalities involve expressions that are ratios of polynomials set in an inequality format. To solve them, one typically finds the values of the variable that make the inequality true, often requiring the identification of critical points where the expression is undefined or equals zero. Understanding how to manipulate these inequalities is crucial for finding the solution set.

Interval Notation

Interval notation is a mathematical notation used to represent a range of values. It uses parentheses and brackets to indicate whether endpoints are included (closed intervals) or excluded (open intervals). For example, the interval (2, 5] includes all numbers greater than 2 and up to 5, including 5 itself. This notation is essential for expressing the solution set of inequalities clearly.

Critical Points

Critical points are values of the variable where the rational expression is either zero or undefined. These points are essential in solving rational inequalities as they divide the number line into intervals that can be tested to determine where the inequality holds true. Identifying these points allows for a systematic approach to finding the solution set.