LESSON: Solving Inequalities

READ: Graph Inequalities on a Number Line

Graph Inequalities on a Number Line

In the last few lessons you have been learning about equations and about balancing an equation. Let’s think about equations. An equation is a number statement with an equal sign. The equal sign tells us that the quantity on one side of the equation is equal the quantity on the other side of the equation. We can solve an equation by figuring out the quantity that will make the equation a true statement.

Example 1

x+5=12

If we think about this equation, we can use mental math and we know that the unknown quantity is equal to 7. If we substitute 7 in for x, we will have a true statement.

7 + 5 &= 12\\ 12 &= 12

Our equation is balanced because one side is equal to the other side. Notice that there is one answer for x that makes this a true statement.


What is an inequality?

An inequality is a mathematical statement that can be equal, but can also be unequal.

We use the following symbols to show that we are working with an inequality.

> means greater than

< means less than

\ge means greater than or equal to

\le means less than or equal to


How do we apply these symbols?

Well, if you think about it, we use the symbols to make a true statement. Let’s look at an example.


Example 2

x + 3 > 5

There are many possible answers that will make this a true statement. We need the quantity on the left side of the inequality to create a sum that is greater than five. Notice that the sign does not have a line under it. We want a quantity that is greater than five not greater than or equal to five on the left side of the inequality.

To make this true, we can choose a set of numbers that has a 3 or greater in it.

x = \{3, 4, 5 \ldots .\}


We don’t need to worry about solving inequalities yet, the key thing to notice is that there are many possible answers that will make an inequality a true statement.

We can use graphs to help us understand inequalities in a visual way.

Graphing inequalities on a number line can help us understand which numbers are s solutions for the inequality and which numbers are not solutions.

Here are some tips for graphing inequalities on a number line.

  • Use an open circle to show that a value is not a solution for the inequality. You will use open circles to graph inequalities that include the symbols > or < .
  • Use a closed circle to show that a value is a solution for the inequality. You will use closed circles to graph inequalities that include the symbols \ge or \le.

Example 3

Graph this inequality x > 3

To complete this task, first draw a number line from –5 to 5.

The inequality x > 3 is read as “x is greater than 3.” So, the solutions of this inequality include all numbers greater than 3. It does not include 3, so draw an open circle at 3 to show that 3 is not a solution for this inequality. Then draw an arrow showing all numbers greater than 3. The arrow should face right because the greater numbers are to the right on a number line.

The graph above shows the solutions for the inequality x > 3.


The graph above can help you see which numbers are solutions for x > 3 and which are not. For example, the arrow includes the numbers 3.5, 4, and 5. If you continued the number line, the arrow would also include the numbers 10 and 100. So, all those numbers––3.5, 4, 5, 10, and 100––are possible values for x.

Example 4

Graph this inequality x<-1.

First, draw a number line from –5 to 5.

The inequality x<-1 is read as “x is less than -1.” So, the solutions of this inequality include all numbers less than -1. It does not include -1, so draw an open circle at -1 to show that -1 is not a solution for this inequality. Then draw an arrow showing all numbers less than -1. The arrow should face left because the lesser numbers are to the left on a number line.

The graph above shows the solutions for the inequality x<-1.


Example 5

Graph this inequality x \ge 0.

First, draw a number line from –5 to 5.

The inequality x \ge 0 is read as “x is greater than or equal to 0.” So, the solutions of this inequality include zero and all numbers that are greater than 0. Draw a closed circle at 0 to show that 0 is a solution for this inequality. Then draw an arrow showing all numbers greater than 0.

The graph above shows the solutions for the inequality x \ge 0.