Solve each rational inequality. Give the solution set in interval notation. See Examples 4 and 5. 8 /(x - 2) ≥ 2

Verified step by step guidance

1. Rewrite the inequality: \( \frac{8}{x - 2} \geq 2 \).
2. Subtract 2 from both sides to set the inequality to zero: \( \frac{8}{x - 2} - 2 \geq 0 \).
3. Combine the terms on the left side over a common denominator: \( \frac{8 - 2(x - 2)}{x - 2} \geq 0 \).
4. Simplify the numerator: \( \frac{8 - 2x + 4}{x - 2} \geq 0 \) becomes \( \frac{12 - 2x}{x - 2} \geq 0 \).
5. Find the critical points by setting the numerator and denominator to zero: Solve \( 12 - 2x = 0 \) and \( x - 2 = 0 \).

Key Concepts

Rational Inequalities

Rational inequalities involve expressions that are ratios of polynomials set in relation to an inequality (e.g., ≥, ≤, >, <). To solve these inequalities, one must determine where the rational expression is positive or negative, which often requires finding critical points where the expression equals zero or is undefined.

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 interval) or excluded (open interval). For example, (a, b) means all numbers between a and b, not including a and b, while [a, b] includes both endpoints.

Critical Points

Critical points are values of the variable that make the rational expression equal to zero or undefined. These points are essential for determining the intervals to test in the inequality. In the context of the given inequality, finding critical points helps to establish the intervals where the inequality holds true.