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 separately: solve \( x - 3 = 0 \) and \( x + 5 = 0 \). These points divide the number line into intervals.
2. Determine the intervals based on the critical points found: \( (-\infty, -5) \), \( (-5, 3) \), and \( (3, \infty) \). Note that \( x = -5 \) is excluded because it makes the denominator zero.
3. Test a sample value from each interval in the inequality \( \frac{x - 3}{x + 5} \leq 0 \) to check whether the expression is less than or equal to zero in that interval.
4. Include the points where the expression equals zero in the solution set. Since the numerator \( x - 3 = 0 \) at \( x = 3 \), check if this point satisfies the inequality and include it if it does.
5. Combine the intervals where the inequality holds true, excluding any points that make the denominator zero, and express the solution set in interval notation.

Key Concepts

Rational Inequalities

Rational inequalities involve expressions where one polynomial is divided by another, and the inequality compares this ratio to zero or another value. Solving them requires understanding where the expression is positive, negative, or zero by analyzing the numerator and denominator separately.

Critical Points and Sign Analysis

Critical points are values that make the numerator or denominator zero, dividing the number line into intervals. By testing values in each interval, you determine the sign of the rational expression, which helps identify where the inequality holds true.

Interval Notation and Domain Restrictions

Interval notation expresses solution sets compactly using parentheses and brackets to indicate open or closed intervals. Domain restrictions exclude values that make the denominator zero, ensuring the solution set only includes valid inputs for the rational expression.