Solve each rational inequality. Give the solution set in interval notation. (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 domain of the inequality by excluding values that make the denominator zero. In this case, \(x \neq -5\) because it makes the denominator zero and the expression undefined.
3. Test a value from each interval created by the critical points in the inequality \(\frac{x - 3}{x + 5} \leq 0\) to determine where the expression is less than or equal to zero.
4. Include the points where the numerator is zero (since the inequality is \(\leq 0\)) but exclude points where the denominator is zero, as the expression is undefined there.
5. Write the solution set in interval notation by combining the intervals where the inequality holds true, making sure to use brackets \([\,]\) for included endpoints and parentheses \((\, )\) for excluded endpoints.

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 finding where the expression is positive, negative, or zero, considering the domain restrictions where the denominator is not zero.

Critical Points and Sign Analysis

Critical points are values that make the numerator or denominator zero, dividing the number line into intervals. By testing points 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. When solving rational inequalities, it's essential to exclude values that make the denominator zero, as these are not in the domain and cannot be part of the solution.