Solve each rational inequality. Give the solution set in interval notation. (2x + 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 \(2x + 3 = 0\) and \(x - 5 = 0\) to find values where the expression is zero or undefined.
2. Determine the critical points: \(x = -\frac{3}{2}\) from the numerator and \(x = 5\) from the denominator. These points divide the number line into intervals to test.
3. Set up intervals based on the critical points: \(( -\infty, -\frac{3}{2} )\), \( ( -\frac{3}{2}, 5 )\), and \( (5, \infty )\).
4. Choose a test value from each interval and substitute it into the expression \(\frac{2x + 3}{x - 5}\) to determine the sign (positive or negative) of the expression in that interval.
5. Based on the inequality \(\leq 0\), select intervals where the expression is negative or zero. Remember to exclude \(x = 5\) because the expression is undefined there. Express the solution set in interval notation accordingly.

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.