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 - 2)/(x + 2) ≤ 2

Verified step by step guidance

1. Start by rewriting the inequality to have zero on one side: \( \frac{(x - 2)}{(x + 2)} \leq 2 \) becomes \( \frac{(x - 2)}{(x + 2)} - 2 \leq 0 \).
2. Find a common denominator to combine the terms on the left side: \( \frac{(x - 2)}{(x + 2)} - \frac{2(x + 2)}{(x + 2)} \leq 0 \).
3. Simplify the numerator: \( \frac{(x - 2) - 2(x + 2)}{(x + 2)} \leq 0 \), which simplifies to \( \frac{x - 2 - 2x - 4}{(x + 2)} \leq 0 \).
4. Combine like terms in the numerator: \( \frac{-x - 6}{(x + 2)} \leq 0 \).
5. Determine the critical points by setting numerator and denominator equal to zero: numerator \( -x - 6 = 0 \) and denominator \( x + 2 = 0 \). Use these points to test intervals on the number line and find where the inequality holds true. Express the solution set in interval notation and graph it accordingly.

Key Concepts

Rational Inequalities

Rational inequalities involve expressions where one polynomial is divided by another, and the inequality compares this ratio to a number or another expression. Solving them requires finding values of the variable that make the inequality true, considering where the expression is defined and the sign of the numerator and denominator.

Critical Points and Sign Analysis

Critical points are values where the numerator or denominator equals zero, dividing the number line into intervals. By testing points in each interval, you determine the sign of the rational expression, which helps identify where the inequality holds true.

Interval Notation and Graphing Solutions

After solving the inequality, the solution set is expressed using interval notation, which concisely represents all values satisfying the inequality. Graphing on a number line visually shows these intervals, including open or closed points depending on whether endpoints are included.