Solve each rational inequality. Give the solution set in interval notation. See Examples 8 and 9. (x+1)/(x-4)>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 \(x - 4 = 0\). These points divide the number line into intervals.
2. The critical points are \(x = -1\) and \(x = 4\). Use these points to create intervals: \(( -\infty, -1 )\), \((-1, 4)\), and \((4, \infty)\).
3. Determine the sign of the rational expression \(\frac{x+1}{x-4}\) on each interval by choosing a test point from each interval and substituting it into the expression.
4. Analyze the inequality \(\frac{x+1}{x-4} > 0\) to decide which intervals satisfy the inequality (where the expression is positive). Remember to exclude points where the denominator is zero, as the expression is undefined there.
5. Write the solution set in interval notation, including only the intervals where the inequality holds true and excluding any points where the expression is undefined.

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 values from each interval in the inequality, you determine where the expression is positive or negative, which helps identify the solution set.

Interval Notation

Interval notation is a concise way to represent sets of real numbers, using parentheses for excluded endpoints and brackets for included ones. It is used to express the solution set of inequalities clearly, especially when describing ranges where the inequality holds true.