Solve each rational inequality. Give the solution set in interval notation. See Examples 4 and 5. 20/(x - 1) ≥ 1

Verified step by step guidance

1. Rewrite the inequality: \( \frac{20}{x - 1} \geq 1 \).
2. Subtract 1 from both sides to set the inequality to zero: \( \frac{20}{x - 1} - 1 \geq 0 \).
3. Combine the terms on the left side over a common denominator: \( \frac{20 - (x - 1)}{x - 1} \geq 0 \).
4. Simplify the numerator: \( \frac{21 - x}{x - 1} \geq 0 \).
5. Determine the critical points by setting the numerator and denominator equal to zero: solve \( 21 - x = 0 \) and \( x - 1 = 0 \).

Key Concepts

Rational Inequalities

Rational inequalities involve expressions that contain a ratio of polynomials, where one side of the inequality is a rational function. To solve these inequalities, we typically find the critical points by setting the numerator and denominator to zero, which helps identify intervals to test for the inequality's validity.

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, (a, b) means all numbers between a and b, not including a and b, while [a, b] includes both endpoints.

Test Intervals

After identifying critical points from a rational inequality, we divide the number line into intervals and select test points from each interval to determine where the inequality holds true. By substituting these test points into the original inequality, we can ascertain which intervals satisfy the condition, leading to the final solution set.