Solve each rational inequality. Give the solution set in interval notation. See Examples 4 and 5. 2 /{x - 2} ≥ 1 / x

Verified step by step guidance

1. Step 1: Start by moving all terms to one side of the inequality to set it to zero: \( \frac{2}{x - 2} - \frac{1}{x} \geq 0 \).
2. Step 2: Find a common denominator for the fractions, which is \( x(x - 2) \). Rewrite each fraction with this common denominator: \( \frac{2x}{x(x - 2)} - \frac{x - 2}{x(x - 2)} \geq 0 \).
3. Step 3: Combine the fractions into a single fraction: \( \frac{2x - (x - 2)}{x(x - 2)} \geq 0 \). Simplify the numerator: \( \frac{2x - x + 2}{x(x - 2)} \geq 0 \).
4. Step 4: Simplify the expression further: \( \frac{x + 2}{x(x - 2)} \geq 0 \).
5. Step 5: Determine the critical points by setting the numerator and denominator equal to zero: \( x + 2 = 0 \) and \( x(x - 2) = 0 \). Solve these equations to find the critical points and test intervals around these points to determine where the inequality holds.

Key Concepts

Rational Inequalities

Rational inequalities involve expressions that contain rational functions, which are ratios of polynomials. To solve these inequalities, one must determine where the rational expression is greater than, less than, or equal to a certain value. This often requires finding critical points where the expression is undefined or equals zero, and then testing intervals to establish 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 but not 2. Understanding how to express solutions in this format is essential for conveying the results of inequalities.

Critical Points

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