Lesson 7.6: Solve Rational Inequalities

New Concept

We're moving beyond equations to solve rational inequalities. This involves finding critical points where the expression is zero or undefined, testing intervals on a number line, and expressing the solution using interval notation.

What’s next

Next, you'll walk through guided examples to see this method in action. Then, you'll apply your skills to a series of practice problems.

Property

A rational inequality is an inequality that contains a rational expression. Inequalities such as $\frac{3}{2x} > 1$, $\frac{2x-3}{x-3} < 4$, $\frac{2x-3}{x-6} \geq x$, and $\frac{1}{4} - \frac{2}{x^2} \leq \frac{3}{x}$ are rational inequalities as they each contain a rational expression.

Examples

• An example of a simple rational inequality is $\frac{x+5}{x-1} > 0$.

• A more complex one could involve a quadratic expression, such as $\frac{3}{y^2-4} \leq 1$.

Property

A critical point is a number which makes the rational expression zero or undefined.

How to Solve a Rational Inequality

1. Write the inequality as one quotient on the left and zero on the right.
2. Determine the critical points—the points where the rational expression will be zero or undefined.
3. Use the critical points to divide the number line into intervals.
4. Test a value in each interval to determine the sign of the quotient.
5. Determine the intervals where the inequality is correct. Write the solution in interval notation.

Examples

• To solve $\frac{x-2}{x+4} \geq 0$, find critical points $x=2$ and $x=-4$. Testing the intervals $(-\infty, -4)$, $(-4, 2)$, and $(2, \infty)$ shows the solution is $(-\infty, -4) \cup [2, \infty)$.

Property

When working with rational functions, it is sometimes useful to know when the function is greater than or less than a particular value. This leads to a rational inequality. For example, to find the values of $x$ that make the function $R(x) = \frac{x+3}{x-5}$ less than or equal to $0$, we set up the inequality $R(x) \leq 0$ and substitute the expression for $R(x)$: $\frac{x+3}{x-5} \leq 0$.

Examples

• Given the function $R(x) = \frac{x-4}{x+1}$, find the values of $x$ that make the function greater than or equal to $0$. The inequality is $\frac{x-4}{x+1} \geq 0$. The solution is $(-\infty, -1) \cup [4, \infty)$.

• For the function $f(x) = \frac{x}{x-3}$, find the values of $x$ where $f(x) < 0$. The critical points are $x=0$ and $x=3$. The function is negative on the interval $(0, 3)$.