A rational inequality is similar to a rational equation—it contains one or more rational expressions—but instead of an equal sign, it includes an inequality symbol like [latex]<[/latex], [latex]\leq[/latex], [latex]>[/latex], or [latex]\geq[/latex].
rational inequality
A rational inequality is an inequality that contains at least one rational expression where the variable appears in a denominator.
Solve the rational inequality:[latex]\dfrac{x - 4}{x + 2} > 0[/latex]
Step 1: Identify critical values The numerator is zero when [latex]x = 4[/latex] The denominator is zero when [latex]x = -2[/latex] (excluded from the solution) These values divide the number line into intervals:
Interval 1: [latex]x < -2[/latex]
Interval 2: [latex]-2 < x < 4[/latex]
Interval 3: [latex]x > 4[/latex]
Step 2: Test a point from each interval
Interval
Test Point
Sign of [latex]\frac{x - 4}{x + 2}[/latex]
[latex]x < -2[/latex]
[latex]-3[/latex]
[latex]\frac{-7}{-1} = 7[/latex] → positive
[latex]-2 < x < 4[/latex]
[latex]0[/latex]
[latex]\frac{-4}{2} = -2[/latex] → negative
[latex]x > 4[/latex]
[latex]5[/latex]
[latex]\frac{1}{7}[/latex] → positive
We are looking for where the expression is greater than 0, so we want the intervals where the expression is positive.
Step 3: Write the solution The solution is: [latex](-\infty, -2) \cup (4, \infty)[/latex]
Note: We use open intervals because the expression is strictly greater than 0 [latex]x = -2[/latex] makes the expression undefined [latex]x = 4[/latex] makes the numerator zero, which gives 0—not greater than 0
To solve a rational inequality:
Write the inequality so that one side is zero and the other side is a single rational expression
Identify critical values (zeros of the numerator and undefined points from the denominator)
Use a sign chart to test the sign of the expression in each interval
Determine the interval(s) where the expression satisfies the inequality
Common Mistakes When Solving Rational Inequalities
Including excluded values in the solution:
Always identify and exclude values that make the denominator zero.
Not setting one side to zero before testing signs:
The sign chart method only works when the inequality is compared to zero.
Forgetting open vs. closed intervals:
If the inequality is strict ([latex]<[/latex] or [latex]>[/latex]), use open intervals for all values. If it includes equality ([latex]\leq[/latex] or [latex]\geq[/latex]), to include values that make the numerator zero (but still exclude points that make the denominator zero).
Solve the rational inequality:
[latex]\dfrac{x + 3}{x - 5} \leq 0[/latex]
Step 1: Identify critical values
Numerator: [latex]x + 3 = 0 \Rightarrow x = -3[/latex]