An inequality is a way to express an unequal relationship between two values. A nonlinear inequality is when one or both sides of the inequality contain a nonlinear expression, such as a quadratic.

**Examples of Nonlinear Inequalities**

An inequality that contains any nonlinear expression is a nonlinear inequality. Examples of nonlinear inequalities include:

While the process of solving polynomial inequalities and rational expression inequalities is similar, the early steps are slightly different, so we will look at them each separately.

**Steps for Solving a Nonlinear Inequality**

- Set the inequality equal to 0.
- Solve for the critical values.
- Graph the critical values on a number line.
- Choose your test points.
- Test the test points.

You will most frequently only see polynomials with the highest power of 2 (e.g. quadratics), but these steps will also work for higher powers. We will go into detail for each step for solving a polynomial inequality.

**1. Set the Inequality to 0 **

The first step to solving a polynomial inequality is to set the inequality to 0, so we can solve for the zeros.

**2. Solve for the critical values**

The critical values of an inequality are where the inequality equals 0. We can solve for these the way we would normally solve a quadratic equation: by factoring and setting the factors equal to zero.

**3. Graph the critical values on a number line**

The next step is to graph the critical values on a number line, to create our test sections.

This creates three test sections, as seen above.

**4. Choose your test points**

To choose your test points, choose any point that lies within the test sections.

We will be using *x *= -3, *x *= 0, and *x *= 3 as the test points, but you are free to use any number that falls within the bounds of the critical points.

**5. Test the test points**

We now plug the test points into the equation to see if they are true or false. We can plug them into the original equation, or the equation with one side equal to 0, from step 1.

We can plot our results visually on the number line:

The inequality is true when x<-2 and where x>2. This can be written as

* x *< -2 and *x *> 2, or (-∞,-2) ∪ (2, ∞) in interval notation.

**Graphing a Nonlinear Inequality**

- To graph a nonlinear inequality, graph the equation and shade in the areas that tested “true” in step 5.
- Draw the graph with a dotted line for less than and greater than ( < and >), and with a solid line for less than or equal to and greater than or equal to (≤ and ≥)

**Example**

A rational inequality involves an expression where there are one or more x values in the denominator. We can follow the same steps of solving a polynomial inequality to solve a rational inequality, but the process of finding the critical values will be different.

**1. Set the inequality to 0**

Since one side of the inequality is already set to 0, we do not need to do anything else in this step.

**2. Solve for the critical values**

The critical values are where the equation equals zero, or is undefined. To find the critical values of a rational inequality, we must find where the numerator and denominator equal 0. To do this, we set the numerator and denominator separately equal to 0, and factor if necessary.

The critical values are *x* = -3, -1, 2.

**3. Graph the critical values on a number line**

From this point, the steps to solving a rational inequality are the same as for solving for a polynomial inequality.

**4. Choose your test points**

Choose test points that lie between the bounds of the critical points. Make sure to choose a test point in all of the bounds!

**5. Test the test points**

Rational inequalities may have more points to test than polynomial inequalities, so make sure to test them all!

The solution is −3 < đť‘Ą < −1 and đť‘Ą > 2, or (−3, −1) ∪ (2, ∞) in interval notation.

**Graph**

Below is a graph of the previous example. Although you won’t be expected to graph complicated rational inequalities, seeing the graph might be helpful to visualize the inequality.

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