Inequalities are exactly like they sound, equations where the sides are "inequal" (not equal) to each other. There are five basic inequalities that you need to be familiar with:
|≤||less than or equal to|
|≥||greater than or equal to|
|≠||not equal to|
The inequality y ≤ 2 means that y can be a number less than 2 (1.9, ¾, 0, -6, etc…) or it can be equal to 2.
How do you remember which one is which? Less than and greater than can easily become mixed up, so we like to think of them as an incomplete Pac Man (or, if you prefer, Ms. Pac Man). Pac Man, being the hungry circle he is, always wants to eat the bigger number, so his "mouth" will be open towards the larger number.
How to graph inequalities
- Draw a circle around the number to which the variable is unequal.
- Fill in the circle if and only if the variable can also equal that number.
- Shade all numbers the variable can be.
Here is what y ≤ 2 looks like:
Here is what y < 2 looks like:
Notice the subtle difference between the two graphs. In the first graph the circle around the 2 is colored in. This is because y can be 2 in the first, but not the second.
Graphing Inequalities Examples
j > -3.5
In this example, the circle around the -3.5 is not colored in and all numbers to the right of the circle are shaded. This is because -3.5 is less than j; or we could say that j is greater than -3.5.
e ≠ ¾
Here the variable can be any number besides ¾, so we need to shade in everything that is not ¾.
-10 ≥ x
The circle is colored in because x can be -10 and x can be smaller than -10, so we shade all numbers to the left.
Look Out: if you switch the terms on each side of the inequality, be very careful to change the sign, too. For example, x > 6 is the same as 6 < x.
Compound inequalities are two or more inequalities combined in the same statement. They often include the words "and" or "or". With "and inequalities," you only graph the numbers that satisfy both inequalities. With "or inequalities," you graph the numbers that satisfy either inequality, or both at the same time.
Let's start by looking at an "or" example in depth.
y > -1 or y ≤ -3
If we break this apart it is two separate inequalities:
y > -1
y ≤ -3
For an "or" inequality we combine all possible values of x onto one number line:
Now let's look at an "and" inequality:
-0.5 < z and z ≤ ¼
(This can and should be combined and written as -0.5 < z ≤ ¼)
-0.5 < z
z ≤ ¼
Now, with "and" we only graph the numbers that satisfy both conditions; i.e. the numbers greater than -0.5 and less than or equal to ¼.