This section covers:

**Graphing Quadratic Inequality Functions****Solving Quadratic Inequalities****Graphing Calculator Method****Completing the Square Method****Sign Chart (Sign Pattern) Method****Real World Quadratic Inequality****More Practice**

Just like we solved and graphed **Linear Inequalities**, we can do the same with **Quadratic Inequalities.**

# 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 “**y <** ”, we always shade in **under **the line that we draw, and when we have “**y >** ”, we always shade **above** the line that we draw. We can even do these on a graphing calculator!

Again, 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 really including that line (but everything up to it), whereas with “” and “”, 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 “” and “”, 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 “” and “” means you draw a solid line on the graph.

# 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 or , or as complicated as or . We’ll use these as examples below.

Remember if you have a **negative** coefficient of , 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.

## Graphing Calculator Method

You probably won’t be able to use a Graphing Calculator on a test, but it’s a good way to check your answers. What we’ll do is put in the left part of the equation in and the right part in , see where they cross, and check which intervals are either greater than or less than, depending on the problem.

Here are some examples:

## Completing the Square Method

We learned how to **Complete the Square **here so we could use the square root method to solve a **quadratic equation** earlier.

We’ll do the same thing here – but when we take the square root of each side, we’ll need to worry about the inequality sign. We still need to break the equation into two equations like we did earlier(one with a plus, one with a minus), but **the equation with the minus must have an inequality sign change.**

For example, if we have , we are saying the absolute value of x is less than the square root of 4, or .

But all you have to remember is when you take the square root of both sides, divide up the equation into two equations.

We get the first equation by just taking away the absolute value sign away on the left. The easiest way to get the second equation is to take the absolute value sign away on the left, and do two things on the right**: reverse the inequality sign**, and **change the sign of everything on the right** (even if we have variables over there). Then solve both and put together (the “or”) in interval notation.

It’s like when we solved **Absolute Value and Inequalities **in the **Linear Inequalities** section.

Here is an example:

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

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

OK, so I’d love to introduce you to the **sign chart **or** sign pattern method** – a method that you’ll use later in Algebra when you work with **Rational Inequalities** in the** Graphing Rational Functions, including Asymptotes** section, and on into Calculus! It looks difficult at first, but really isn’t too bad at all!

A **sign chart** or **sign pattern** is simply a number line that is separated into partitions (or intervals or regions), with boundary points (called “**critical values**“) that you get by setting the quadratic to 0 (without the inequality) and solving for ** x** (the roots).

Sign charts are easy and a lot of fun since you can pick any point in between the **critical values**, and see if the whole quadratic is **positive** or **negative**. Then you just pick that interval (or intervals) by looking at the inequality.

Also, 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**).

Let’s do some examples:

One thing to 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.

Here is another example:

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

# Quadratic Inequality Real World Example

Supposed Aven drops a ball off the top of a 10 foot pool slide, and the ball follows the projectile , where ** t** is the time in seconds, and

**is the height of the ball. Her friend Riley needs to catch the ball between 2 feet and 5 feet off the top of the water (ground). Between what two times should Riley try to catch the ball?**

*h***Solution:**

Since we need to know when (** t**) the ball should be caught, we need to solve for

**, use an inequality since it can be anytime between 2 and 5 seconds. We’ll use just < signs since the problem says “between” and not “inclusive”. So let’s solve:**

*t*

**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!

thanks a lot for your information. if you dont mind, could you please help me to find some real application of inequalities quadratic in real life? i need it so much. thanks a lot

Thanks so much for writing. I thought this was a good thing to add, so I added it here. Hope it makes sense. Lisa

Here’s a wonderful article by Nghi H. Nguyen on the Best Method for Solving Quadratic Inequalities in One Variable:

http://www.shelovesmath.com/wp-content/uploads/2014/12/Best-Method-For-Solving-Quadratic-Inequalities-in-one-variable.pdf

Hi my name is Hannah and I need some help trying to get a quadratic Word problem

Can you give me an example of what you need? I have some quadratic word problems

here, if you want. Thanks, LisaHii.. Its me pranav.

I have problem in solving inequalities having quadratic equations under absolute value function. Could you please give some examples for the same..

Thanks for writing! I do need to add a section on this – thanks! I do have a problem somewhat like this in the Piecewise Function section –

HERE– does that help? Lisa