Graphing One-Variable Inequalities
By S Taylor
When we solve an equation and come up with a solution such as 
, that means that 4 is the only value that makes that equation true. But when solving inequalities, we get an answer that is also in the form on an inequality 
. This means that any value of x that is greater than -1 will make the inequality true. In other words, there are infinitely many values that will work. If you want to see how linear inequalities are solved, click here.
This lesson will focus on how to graph the solution to an inequality. Let’s go back and look at the inequality mentioned above,
. To show that all values greater than -1 are part of the solution, we can draw a number line and graph the solution. Let’s start by drawing a blank number line. It can be as long or as short as you like as long as it shows the solution completely.
Now we need to draw our solution on the number line.
Since -1 is not included in the solution, we do not fill in the circle. Whenever the inequality is < or > there will be an open circle on the number line. Since every number larger than -1 is included, we want to shade that portion of the number line.
If we had an inequality like
we would shade the circle at 2 and shade the number line to the left of 2. (Less than means we shade to the Left.)
So far we’ve see that symbols of < and > mean you have an open circle and symbols of
and 
you have filled in circle. When > and 
are used, you shade to the right. And when < and 
are used, you shade to the left.
We can use these same rules for double inequalities. Remember that a double inequality “sandwiches” the variable between two values, like
. This means any values of x between -1 and 3 are part of our solution. In this case, our number line graph will not have an end portion shaded, but rather just be shaded between -1 and 3.
It is possible for the two inequalities in a double inequality to be different. In other words, one side may have a filled in circle yet the other side can be an open circle. Look at the graph for
, that means that 4 is the only value that makes that equation true. But when solving inequalities, we get an answer that is also in the form on an inequality . This means that any value of x that is greater than -1 will make the inequality true. In other words, there are infinitely many values that will work. If you want to see how linear inequalities are solved, click here.
This lesson will focus on how to graph the solution to an inequality. Let’s go back and look at the inequality mentioned above,
. To show that all values greater than -1 are part of the solution, we can draw a number line and graph the solution. Let’s start by drawing a blank number line. It can be as long or as short as you like as long as it shows the solution completely.
Now we need to draw our solution on the number line.
Since -1 is not included in the solution, we do not fill in the circle. Whenever the inequality is < or > there will be an open circle on the number line. Since every number larger than -1 is included, we want to shade that portion of the number line.
If we had an inequality like
we would shade the circle at 2 and shade the number line to the left of 2. (Less than means we shade to the Left.)
So far we’ve see that symbols of < and > mean you have an open circle and symbols of
and you have filled in circle. When > and are used, you shade to the right. And when < and are used, you shade to the left.
We can use these same rules for double inequalities. Remember that a double inequality “sandwiches” the variable between two values, like
. This means any values of x between -1 and 3 are part of our solution. In this case, our number line graph will not have an end portion shaded, but rather just be shaded between -1 and 3.
It is possible for the two inequalities in a double inequality to be different. In other words, one side may have a filled in circle yet the other side can be an open circle. Look at the graph for
Graph the solution to each of these inequalities from the lesson on solving linear inequalities.
1.
2.
3.
4.
5.
6.
