Solve each rational inequality. Give the solution set in interval notation. See Examples 8 and 9. (x-3)/(x+5)≤0

Key Concepts

Rational Inequalities

Rational inequalities involve expressions that are ratios of polynomials set in relation to an inequality (e.g., ≤, ≥). To solve them, one must determine where the rational expression is positive, negative, or zero. This often requires finding critical points where the numerator or denominator equals zero and testing intervals around these points.

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 where the rational expression is either zero or undefined. For the inequality (x-3)/(x+5)≤0, the critical points are found by setting the numerator (x-3) to zero and the denominator (x+5) to zero. These points help to divide the number line into intervals for testing the sign of the rational expression.