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

Verified step by step guidance

1. Identify the critical points by setting the numerator and denominator equal to zero: \(x - 3 = 0\) and \(x + 5 = 0\).
2. Solve these equations to find the critical points: \(x = 3\) and \(x = -5\).
3. Use these critical points to divide the number line into intervals: \((-\infty, -5)\), \((-5, 3)\), and \((3, \infty)\).
4. Test a point from each interval in the inequality \(\frac{x-3}{x+5} \leq 0\) to determine where the inequality holds true.
5. Consider the critical points: check if they satisfy the inequality and include them in the solution set if they do. Finally, express the solution set in interval notation.

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.