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Teachers & SchoolsStudy Guide

Inequalities are exactly what they sound like: equations where the sides are "inequal" (not equal) to each other. There are five basic inequalities that we need to be familiar with:

Symbol | Meaning |

< | less than |

> | greater than |

≤ | less than or equal to |

≥ | greater than or equal to |

≠ | not equal to |

The inequality *y* < 2 means that *y* can be any number less than 2 (such as 1.9, 0.75, 0, -6, etc.).

The inequality *y* > 7 means that *y* can be any number greater than 7 (such as 7.1, 8, 9, 537, etc.).

The inequality *y* ≤ 2 means that *y* can be any number less than 2, *or* it can be equal to 2 itself (2, 1.9, 1, 0, -6, etc.).

Last but not least, the inequality *y* ≥ 7 means that *y* can be any number greater than 7, *or* it can be equal to 7 (7, 7.001, 8, 9, 200, etc.). That little line underneath an inequality symbol means "or equal to."

How do we remember which one is which? "Less than" and "greater than" are easy to mix 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 always be open towards the larger number.

We can graph inequalities on a number line to get a better idea of how they're behaving. Just follow these steps.

- Find the number on the other side of the inequality sign from the variable (like the 4 in
*x*> 4). - Sketch a number line and draw an open circle around that number.
- Fill in the circle if and only if the variable
*can*also equal that number. - Shade all numbers the variable can be.

Here's what *y* ≤ 2 looks like:

Here's 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.

**Example 1**

*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.

**Example 2**

*p* ≠ ¾

Here the variable can be any number besides ¾, so we need to shade in everything that's not ¾.

**Example 3**

-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, we only graph the numbers that satisfy both inequalities, a.k.a. the **intersection** of both inequalities. With "or" inequalities, we graph the numbers that satisfy either inequality, or both at the same time. In other words, we graph the combination, or **union**, of both inequalities.

Let's start by looking at an "or" example in depth.

*y* > -1 or *y* ≤ -3

If we break this apart, we can think of it as two separate inequalities:

*y* > -1

*y* ≤ -3

For an "or" inequality we combine all possible values of *y* onto one number line:

Now let's look at an "and" inequality:

-0.5 < *z* and *z* ≤ 0.25

For starters, we can combine forces and write the inequality like this:

-0.5 < *z* ≤ 0.25

Now tackle each side separately.

-0.5 < *z*

*z* ≤ 0.25

To finish up, we only graph the numbers that satisfy both conditions; i.e. the numbers greater than -0.5 *and* less than or equal to 0.25.