Just like we solved and graphed **Linear Inequalities**, we can do the same with **Quadratic Inequalities. **There are also more types of inequalities that can be found here in the **Solving Inequalities** section.

## Graphing Quadratic Inequality Functions

We learned how to graph inequalities with two variables way back in the **Coordinate System and Graphing Lines** section. We can do the same with quadratics and the shading is pretty much the same: when we have (positive) “$ y<$ ”, we always shade in **under **the line that we draw, and when we have (positive) “$ y>$ ”, we always shade **above** the line that we draw. We can also see these on a graphing calculator.

We can always plug in an ordered pair to see if it shows up in the shaded areas (which means it’s a solution), or the unshaded areas (which means it’s not a solution.) With “$ <$” and “$ >$” inequalities, we draw a **dashed (or dotted) line** to indicate that we’re not including that line (but everything up to it), whereas with “$ \le $” and “$ \ge $”, we draw a **regular line**, to indicate that we are including it in the solution. To remember this, I think about the fact that “$ <$” and “$ >$” have less pencil marks than “$ \le $” and “$ \ge $”, so there is less pencil used when you draw the lines on the graph. You can also remember this by thinking the line under the “$ \le $” and “$ \ge $” means you draw a solid line on the graph. Here is an example:

## Solving Quadratic Inequalities

You will also have to know how to solve quadratic inequalities, which make things a little messy. Examples of **Quadratic Inequalities **can be as “simple” as $ {{x}^{2}}>4$ or $ 4{{x}^{2}}\le 28x$, or as complicated as $ 2{{x}^{2}}-7x\le -3$ or $ \displaystyle {{x}^{2}}+5x-9<0$. We’ll use these as examples below.

Remember if you have a **negative** coefficient of $ \boldsymbol{{{x}^{2}}}$, you can move everything to the other side to make it positive – but be careful of the inequality signs!

There are three main methods used to solve **Quadratic Inequalities.** The **Sign Pattern or Sign Chart Method** is the most preferred, but I’ll cover a couple of methods here first.

## Solving Using Graphing

You can solve quadratic inequalities by graphing the two sides of an inequality and seeing what the $ x$-intervals are for where one graph lies either below ($ <$) or above ($ >$) the other one.

Here are some examples using a **TI Graphing Calculator**. Put the left part of the equation in $ {{Y}_{1}}=$ and the right part in $ {{Y}_{2}}=$, see where they cross, and check which intervals are either greater than or less than, depending on the problem:

## Solving Algebraically, including Completing the Square

When solving algebraically, we can take the square root of each side, but we need to pay attention to the inequality sign. We need to break the equation into two equations like we did in the **Solving Absolute Value Inequalities** section (one with a plus, one with a minus), but **the equation with the minus must have an inequality sign change. **The reason we’re breaking up the inequality into two equations is a square root may be either **positive** or **negative**.

After solving both inequalities, put the solutions together in interval notation. **Remember that with >, we have an “or” and with < we have an “and”.**

Here is an example:

Here is the example where we need to** Complete the Square** first:

## Sign Chart (Sign Pattern) Method – the Easiest Method!

The **sign chart **or** sign pattern method** is used quite frequently in Algebra, including here in **Solving Polynomial Inequalities** from the **Graphing and Finding Roots of Polynomial Functions **section, and here in **Rational Inequalities** from the** Graphing Rational Functions, including Asymptotes** section. It looks difficult at first, but really isn’t too bad at all!

A **sign chart** or **sign pattern** is a number line that is separated into partitions, or intervals or regions, with boundary points, that are called “**critical values**“. You obtain the critical values by setting the quadratic to **0** and solving for $ x$ (the roots).

The idea of sign charts is to pick any point in between the **critical values**, and see if the whole quadratic is **positive** or **negative**, and then use that information to get the solution to the inequality.

It’s a good idea to put **open or closed circles **on the **critical values** to remind ourselves if we have** ****inclusive** points (inequalities with equal signs, such as $ \le $ and $ \ge $) or **exclusive** points (inequalities without equal signs, or **factors in the denominators**).

It’s best to show examples:

**Note**: If there are no squares of any of the factors (with variables) in the quadratic factored form, the **sign chart will typically be** **alternating minus and plus**, like plus-minus-plus, or minus-plus-minus. If one of the factors is raised to an even factor (like squared), then this indicates a “**bounce**” in the graph, and the signs won’t change at that point.

Here are more examples:

And one more example, where we can’t factor:

## Real World Quadratic Inequality Example

Here’s a “real-world” quadratic inequality problem:

**Learn these rules, and practice, practice, practice!**

Click on Submit (the arrow to the right of the problem) to solve this problem. You can also type in more problems, or click on the 3 dots in the upper right hand corner to drill down for example problems.

If you click on “Tap to view steps”, you will go to the **Mathway** site, where you can register for the **full version** (steps included) of the software. You can even get math worksheets.

You can also go to the **Mathway** site here, where you can register, or just use the software for free without the detailed solutions. There is even a Mathway App for your mobile device. Enjoy!

On to **Quadratic Applications** – you are ready!