Solve each rational inequality. Give the solution set in interval notation. (x + 1)/(x - 5) > 0

Verified step by step guidance

1. Identify the critical points by setting the numerator and denominator equal to zero separately: solve \( x + 1 = 0 \) and \( x - 5 = 0 \). These points divide the number line into intervals.
2. Determine the intervals based on the critical points found: \( (-\infty, -1) \), \( (-1, 5) \), and \( (5, \infty) \).
3. Test a sample value from each interval in the inequality \( \frac{x + 1}{x - 5} > 0 \) to check whether the expression is positive or negative in that interval.
4. Use the results from the test points to identify which intervals satisfy the inequality \( \frac{x + 1}{x - 5} > 0 \). Remember to exclude any points where the denominator is zero, as the expression is undefined there.
5. Express the solution set in interval notation, combining all intervals where the inequality holds true.

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 undefined by analyzing the numerator and denominator separately.

Critical Points and Sign Analysis

Critical points are values where the numerator or denominator equals 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 or 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.