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/(x + 2) ≥ 2

Verified step by step guidance

1. Start by rewriting the inequality: \(\frac{x}{x + 2} \geq 2\).
2. Bring all terms to one side to have zero on the other side: \(\frac{x}{x + 2} - 2 \geq 0\).
3. Find a common denominator and combine the terms: \(\frac{x - 2(x + 2)}{x + 2} \geq 0\).
4. Simplify the numerator: \(\frac{x - 2x - 4}{x + 2} = \frac{-x - 4}{x + 2} \geq 0\).
5. Determine the critical points by setting numerator and denominator equal to zero: numerator \(-x - 4 = 0\) and denominator \(x + 2 = 0\). These points divide the number line into intervals to test for the inequality.

Key Concepts

Rational Inequalities

Rational inequalities involve expressions where one rational expression is compared to another using inequality symbols. Solving them requires finding values of the variable that make the inequality true, often by analyzing the sign of the numerator and denominator separately.

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 where the rational expression is positive or negative, which helps identify the solution set for the inequality.

Interval Notation and Graphing on a Number Line

Interval notation concisely represents sets of real numbers that satisfy the inequality, using parentheses or brackets to indicate open or closed intervals. Graphing these intervals on a number line visually shows the solution set and helps verify the correctness of the solution.