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So what is an inequality?

Well, the official definition is:

An inequality is a relation that holds between two values when they are different.

What this means is basically this:

An inequality means it is not equal, but since math deals with numbers, it is either greater than, less than, greater than or equal to, or less than or equal to.

For those of you that are a little rusty on the symbols, here’s what it looks like:

< means less than, ≤ means less than or equal to

> means greater than, ≥ means greater than or equal to

An easy way to remember this is to think of Pacman.

Pacman always tries to eat the bigger number because eating is what he does. In this case:

4 < 7

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Great, how do we solve inequalities?

So inequalities are essentially the same as any other equation (even though they are not equations), we try to simplify them the best we can so we know what numbers we can plug into them to make them true.

So essentially we solve them the same way as we solve them the same way we would solve an equation.

This is one of those times when explaining how to solve is harder then just showing you how to solve, so without further ado…..

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Example

Let’s say you are given something like:

2x + 4 < 20

2x + 4 = 20

To simplify this inequality, think of it like an equation.

So now how would you solve this equation?

-4 -4

2x = 16

_______

2 2

x = 8

Now, remember, we didn’t start with an equation though,

we started with an inequality.

So, instead of our answer being x = 8, we should have…..

x < 8

2x < 16

Now let’s do another one just to make sure you understand

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Last Example

Let’s say you are given something like:

7x + 12 > 33

Again, we just solve this inequality like we would an equation.

So, we subtract 12 from both sides

And now we have:

Then we divide by 7 and we get:

But what does that mean?

Well, this means that if we pick any number that is bigger than 3

and plug it in for x, we will get a number that is bigger than 33.

-12 -12

7x > 21

_______�7 7

x > 3

So, let’s try it!

12 > 3

7(12) + 12 > 33

84 + 12 > 33

96 > 33

What if we pick a number that is bigger than 3, say….12?

Well, if we set x = 12, then…..

Is 12 greater than 3?

Sure!

Now we need to test out 12 in our inequality to see if it’s true as well!

So, is 96 greater than 33?

Sure!

So now we know it’s true.

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SOME RULES TO CONSIDER

So inequalities are special in that, they are not your regular expressions.

Yes, we can undo operations on them like we would an equation, but there are a few rules.

Rule #1: When you multiply an inequality by a negative, you must change the sign.

Rule #2: When you divide an inequality by a negative, you must change the sign.

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Example

Let’s say you are given something like:

-3x - 7 < 14

Well, we know we can simplify this inequality the same way we’ve been doing it, so:

+7 +7

_______

-3 -3

x < -7

-3x < 21

However, we did divide by a negative right?

x > -7

So we should make sure to switch the direction of our inequality.

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DIRECTION OF THE ARROW AND PLACEMENT OF THE CIRCLE

So, when it comes to graphing a one variable inequality, we need to make sure to place two things, a circle and an arrow.

Your circle goes over the number you found when you simplified the inequality, and the arrow goes in the direction that makes it true.

Here’s an example

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Example

Simplify and graph the inequality:

2x + 3 > 23

- 3 - 3

2x > 20

_______� 2 2

x > 10

So first, we need to simplify the inequality

Now we graph it.�

So as we can see, we have: x > 10

This means we need to place our circle over 10 on our graph.

So here’s our graph

Now we need to place our circle on the graph over 10.

Now we need to place our arrow in the direction that makes this true.

Since x is greater than 10, this would mean all the numbers to the right would make it right.

So our graph should look like:

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Example

Let’s say you have something like:

x > 5

Now we have greater than, with no line underneath, so the graph for this would look like:

Now let’s say you have something else like:

Now we have greater than or equal to, as you can see because there is a line underneath. �So the graph for this would look like:

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So, what about inequalities with two variables?

Well, when you deal with inequalities with two variables, you treat them basically the same way you would treat a one variable inequality, by simplifying until you can get y by itself.

It’s basically the same way we try to simplify two variable equations.

This is one of those times where it is harder to explain rather than just show, so without further ado…..

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SOME EXAMPLES OF INEQUALITIES

So, instead of going through and explaining, it may be better to just start off on some examples:

Solve the inequality:

y – 2 > 3x + 4

So, when you want to solve an inequality, the first thing you do is treat it like an equation.

IMPORTANT NOTE!

However, don’t forget the one rule that you need to be aware of.

If you divide (or multiply) an inequality by -1, you need to switch the inequality.

What I mean by that is this:

Example:

-2y < 4x + 10

We know we divide by -2 to get y by itself, however our answer would be:

y > -2x – 5

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SO HOW DO WE GRAPH AN INEQUALITY?

So how do we graph inequalities?

It’s way more simple than you may think.

First we graph what the equation would like.

Next we shade the graph either up, or down depending on whether it is greater than or less than.

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EXAMPLE:

Graph:

Y ≥ 3x + 5

First, we graph the line y = 3x + 5

Then we look and see that the inequality is great than or equal to, so we look at the line and shade up.

And that’s all

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SOME KEY POINTS

Again, if the inequality is y is greater than or equal to, you graph the line and then shade up.
If the inequality is y is less than or equal to, you graph the line and then shade down.

One last point to make:

If the graph is less than, or greater than, you do the same thing BUT MAKE SURE YOU MAKE YOUR LINE DASHED.

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Example:

Graph:

Y > 3x + 5

First, we graph the line y = 3x + 5

Then we look and see that the inequality is great than, so we look at the line and shade up.

And that’s all