Solve each rational inequality. Give the solution set in interval notation. See Examples 8 and 9. (6-x)/(x+2)>1

Verified step by step guidance

1. Start by subtracting 1 from both sides of the inequality: $\frac{6-x}{x+2}-1>0$.
2. Combine the terms on the left side over a common denominator: $\frac{6-x}{x+2}-\frac{x+2}{x+2}>0$.
3. Simplify the expression: $\frac{6-x-(x+2)}{x+2}>0$.
4. Further simplify the numerator: $\frac{6-x-x-2}{x+2}>0$ which becomes $\frac{4-2x}{x+2}>0$.
5. Determine the critical points by setting the numerator and denominator equal to zero: Solve $4-2x=0$ and $x+2=0$ to find the critical points.

Key Concepts

Rational Inequalities

Rational inequalities involve expressions that are ratios of polynomials set in relation to a value, typically using inequality symbols like >, <, ≥, or ≤. 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 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 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.