Solve each rational inequality. Give the solution set in interval notation. See Examples 8 and 9. 4/(2-x)≥3/(1-x)

Verified step by step guidance

1. Start by finding a common denominator for the rational expressions on both sides of the inequality. The common denominator for $\frac{4}{2-x}$ and $\frac{3}{1-x}$ is $(2-x)(1-x)$.
2. Rewrite each fraction with the common denominator: $\frac{4(1-x)}{(2-x)(1-x)}$ and $\frac{3(2-x)}{(2-x)(1-x)}$.
3. Set up the inequality with the new expressions: $\frac{4(1-x)}{(2-x)(1-x)}\ge \frac{3(2-x)}{(2-x)(1-x)}$.
4. Since the denominators are the same, you can compare the numerators: $4(1-x)\ge 3(2-x)$.
5. Solve the inequality $4(1-x)\ge 3(2-x)$ by expanding and simplifying both sides, then isolate $x$ to find the solution set. Remember to consider the values that make the original denominators zero, as these are not included in the solution set.

Key Concepts

Rational Inequalities

Rational inequalities involve expressions that are ratios of polynomials set in relation to each other using inequality symbols (e.g., ≥, ≤, >, <). To solve these inequalities, one must find the values of the variable that make the inequality true, often requiring the identification of critical points where the expression is undefined or equals zero.

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

Critical Points

Critical points are specific values of the variable where the rational expression is either zero or undefined. These points are essential in solving rational inequalities as they divide the number line into intervals. By testing these intervals, one can determine where the inequality holds true, leading to the final solution set.