Solve each rational inequality. Give the solution set in interval notation. (x+1)/(x-4)>0

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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.

Determine the critical points: \(x = -1\) from the numerator and \(x = 4\) from the denominator. Note that \(x = 4\) is a vertical asymptote and cannot be included in the solution.

Divide the number line into three intervals based on the critical points: \(( -\infty, -1 )\), \((-1, 4)\), and \((4, \infty)\).

Test a sample value from each interval in the inequality \(\frac{x+1}{x-4} > 0\) to determine if the expression is positive or negative in that interval.

Based on the sign tests, write the solution set by including intervals where the inequality holds true, and express the solution in interval notation, excluding points where the denominator is zero.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 express solution sets of inequalities using parentheses and brackets to indicate open or closed intervals. It clearly shows the range of values that satisfy the inequality, excluding points where the expression is undefined.

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