6.08 Graph inequalities
Introduction
We've previously looked at how to  graph numbers on a number line . We can also represent solutions to inequalities that include a variable on a number line.
Graph inequalities
Let's look at how we can graph x\leq4 on a number line.
When we say "x is less than or equal to 4", we're not just talking about one number. We're talking about a whole set of numbers, including x=4, x=2, x=0, x=-1 and x=-1000. All of these numbers are less than or equal to 4.
If we graph all of the integers that are less than or equal to 4 on a number line, we get something that looks like this:
Our number line represents all the whole numbers that are shown and less than 4, but what about fractions like \dfrac{1}{2} or decimals like -2.5? These numbers are also less than or equal to 4, so we need to include them in our graph.
Rather than trying to graph all of the individual points, which would get very messy, we can draw a ray that includes all of the points, since they all are numbers that make the inequality true.
What if we instead want to graph the very similar inequality x<4? The only difference now is that x cannot be equal to 4 because 4 is not less than 4, and so the graph should not include the point where x=4.
So we want to graph the same ray, but leave off the point at the end where x=4. To represent this we draw the graph with an unfilled circle, instead of a filled in circle, to show that 4 is not included:
To graph inequalities that are greater than or greater than or equal to, we use the same method but the arrow will be facing to the right. For example, the graph of x>2 looks like this:
Examples
Example 1
State the inequality for x that is represented on the number line.
To graph an inequality, start by determining which direction the ray will point, right or left. This can be determined by making sure that the ray covers all the values make the inequality true.
The end point of the ray will be an unfilled circle if the inequality has a < or >.
The end point of the ray will be a filled circle if the inequality has a \leq or \geq.
Graph solutions to inequalities
Now let's consider an inequality such as x + 3 > 5. What would we graph for this inequality?
We want to graph all of the possible values that the variable can take - which will be all of the solutions that make the inequality true.
The inequality x + 3 > 5 has the solutions "all numbers which, when added to 3 result in a number greater than 5". This is a bit complicated to represent on a number line.
In order to graph the solutions to an inequality, it will be easiest to start with solving the inequality.
| \displaystyle x+3 | \displaystyle > | \displaystyle 5 | Given inequality |
| \displaystyle x+3-3 | \displaystyle > | \displaystyle 5-3 | Subtract 3 from both sides |
| \displaystyle x | \displaystyle > | \displaystyle 2 | Simplify |
We can now graph the solutions of the inequality which are "all numbers greater than 2" on the number line, which looks like this:
Examples
Example 2
Consider the inequality x-4<1
Solve the inequality for x.
graph the solutions to the inequality x-4<1 on the number line beelow.
To graph solutions to inequalities, we start by solving the inequality, followed by graphing the result.