Solve each rational inequality in Exercises 43–60 and graph the solution set on a real number line. Express each solution set in interval notation. (x + 1)/(x + 3) < 2

Verified step by step guidance

1. Step 1: Start by rewriting the inequality \( \frac{x + 1}{x + 3} < 2 \) in a form that can be solved. Subtract 2 from both sides to get \( \frac{x + 1}{x + 3} - 2 < 0 \).
2. Step 2: Combine the terms on the left side over a common denominator. Rewrite 2 as \( \frac{2(x + 3)}{x + 3} \) to get \( \frac{x + 1 - 2(x + 3)}{x + 3} < 0 \).
3. Step 3: Simplify the numerator. Distribute the -2 to get \( x + 1 - 2x - 6 \), which simplifies to \( -x - 5 \). The inequality becomes \( \frac{-x - 5}{x + 3} < 0 \).
4. Step 4: Identify the critical points by setting the numerator and denominator equal to zero. Solve \( -x - 5 = 0 \) to find \( x = -5 \) and \( x + 3 = 0 \) to find \( x = -3 \).
5. Step 5: Use the critical points to test intervals on the number line: \((-\infty, -5)\), \((-5, -3)\), and \((-3, \infty)\). Determine where the inequality holds true and express the solution set in interval notation.

Key Concepts

Rational Inequalities

Rational inequalities involve expressions that are ratios of polynomials set in relation to an inequality (e.g., <, >, ≤, ≥). To solve them, one typically finds critical points by setting the numerator and denominator to zero, which helps identify intervals to test for the inequality's truth. Understanding how to manipulate and analyze these expressions is crucial for finding valid solutions.

Interval Notation

Interval notation is a mathematical notation used to represent a range of values on the real number line. It uses parentheses and brackets to indicate whether endpoints are included (closed intervals) or excluded (open intervals). For example, (a, b) represents all numbers between a and b, not including a and b, while [a, b] includes both endpoints. This notation is essential for clearly expressing the solution set of inequalities.

Graphing Solution Sets

Graphing solution sets on a real number line visually represents the range of values that satisfy the inequality. This involves marking critical points and shading the appropriate regions based on whether the endpoints are included or excluded. Understanding how to accurately graph these solutions helps in interpreting the results and communicating them effectively.