Graphing and Writing Inequalities

Contributor: Mason Smith. Lesson ID: 11232

In life, inequalities may seem unfair. However, in the Math World, inequalities are everywhere (more or less!) and are very important! Learn to use number lines to graph and create inequalities!


Algebra I, Expressions and Equations, Pre-Algebra

learning style
personality style
Grade Level
Middle School (6-8), High School (9-12)
Lesson Type
Quick Query

Lesson Plan - Get It!


Q: Why did the Moore family name their son Lester?

A: So he could either be "Les" or "Moore."

  • What are inequalities?

Inequalities are statements that show that two statements are not equal to one another, unlike equations, where the two sides are equal to each other.

For example, 7 is bigger than 6, so we can use the greater than symbol (>); whereas 9 is less than 10, so we use the less than symbol (

The sign can make a big difference in your solutions, especially when we begin to graph them, so make sure you pay close attention to what the inequality actually says.

Let's practice with another example. If we have x ≥ 2, then what values can x have?

X would be all values x ≥ 2, which means x could be 2, 3, 4, 4.5, 8, all the way up to infinity.

Now that we have a basic understanding of what inequalities are, let's graph them so we can visually see what they mean.

Start by drawing a number line from -4 to 4 on a piece of paper. Then, take the solved equation of b < 1 and graph it using the following method:

  1. First, we find the mentioned value, which is 1, and draw a circle on that number.
  2. Then, look at the inequality sign, and if it has an =, then we color in the circle. If not, leave it as is.
  3. Last, look at what the inequality tells us: since b < 1 means all numbers smaller than 1, shade all numbers with a value less than one on the number line.

When you are done, you should have something like this:

Try the same steps with r ≥ 2.

  1. Find the value and put a circle over it.
  2. If there is a line under the inequality, or an = next to it, then fill in the circle.
  3. Finally, look at what the inequality says, and color or shade the correct side of that value on the number line (> means "greater than," so all the numbers on the right side of the value.)

Your number line should look something like this:

Now that you know how to graph inequalities, you can move on to the last important part of understanding inequalities — being able to turn graphs (or number lines) into inequalities. We will take what we know about graphing inequalities and use that knowledge so we can easily turn any inequality graph into an inequality.

Let's start with this graph:

  1. Look at the shaded direction: this one goes to the right, or up the number line, so it must be greater than; write x >
  2. Is the point shaded? If so, add an = or write ≥. If the point is not shaded, leave the inequality sign alone. We are up to x ≥
  3. What is the point? That will be your numerical value. Here it is –3, so we have x ≥ -3, which is the correct answer.

Now that we have an idea of what inequalities are, let's try some practice!

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