Quadratic Inequalities

This section covers:

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.

Here is an example:

Quadratic Inequality 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:

Solving Quadratic Inequalities with Graphing Calculator

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:

X Squared Less Than 4 Example

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

Quadratic Inequality by Completing the Square

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:

Solving Quadratics Inequality with Sign Chart

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:

Solving Quadratics Inequality with Sign Chart Two

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

Solving Quadratic Inequality with Sign Chart Three

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 Height of Ball, where t is the time in seconds, and h 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?


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

Real World Quadratic Inequality

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!

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