Graphing Linear Inequalities – Explanation and Examples
Graphing linear inequalities is a way of using the coordinate plane to visually show which points satisfy an inequality and which do not.
Graphing linear inequalities is very similar to graphing numerical inequalities. When we have one number, we can use a number line. When we are dealing with two variables, x and y, we can use the Cartesian plane to graph the inequality.
Graphing inequalities requires a thorough understanding of the coordinate plane, the equation of a line, and graphing lines. Make sure to review those topics before moving forward with this one.
In particular, this section will cover:
- How to Graph Inequalities
- Graphing Systems of Inequalities
How to Graph Inequalities
Graphing linear inequalities is a way of visually representing a linear inequality. There are three main steps required to graph a linear inequality.
- Graph the line.
- Decide on a solid or dashed line.
- Shade above or below the line.
Graphing the Line
Recall that a linear equation is a relationship between the independent and dependent variables, usually x and y, that can be modeled as a line in the Cartesian coordinate system. One of the most common linear equations is slope-intercept form, y=mx+b, where m is the slope of the line and b is the y-intercept of the line.
A linear inequality usually looks like a linear equation where the equal sign has been exchanged for a greater than, a less than, a greater than or equal to, or a less than or equal to sign. For example, a linear inequality may look like:
The first step in graphing linear inequalities is graphing the line. That is, if you are given any of the above inequalities, graph the line y=mx+b.
Decide on a Solid or Dashed Line
Now, we need to decide whether the graph of the line y=mx+b should be a solid line or a dashed line. This is similar to deciding whether to have an open circle or a closed circle when graphing a single variable.
That is, if our original linear inequality has a greater than or a less than sign, we use a dashed line. This means the solution to the inequality does not include points that lie on the graphed line.
Alternatively, if the original linear inequality includes a greater than or equal to sign or a less than or equal to sign, we use a solid line. This means the solution to the inequality does include the points that lie on the graphed line.
Shade Above or Below the Line
Finally, we need to decide whether to shade above or below the line we graphed. This is similar to deciding whether to shade to the right or left on a number line when graphing a one-variable inequality.
That is, if the original linear inequality has a greater than or a greater than or equal to sign, then we shade up and to the right of the line. This means the solution to the linear inequality includes points above the graphed line.
Alternatively, if the original linear inequality has a less than or less than or equal to sign, then we shade down and to the left of the line. This means the solution to the linear inequality includes points below the graphed line.
Graphing Systems of Inequalities
Again, just as we can graph systems of inequalities in one variable, we can graph systems of linear inequalities in two variables.
Systems of linear inequalities will be connected by the words AND or OR, and these are often written in all capitals as shown here.
The word “and” in mathematics means both things must be happening. For example, in math, if something is prime and even, only the number two works.
When graphing systems of inequalities connected by the word “and,” we shade the overlap between two or more linear inequalities.
The word “or” in mathematics means “either or both.” The mathematical “or” includes the overlap between two things, whereas every day English does not include both. For example, in math, if something is divisible by 2 or 3, the numbers 4, 6, and 9 all work.
When graphing systems of inequalities connected by the word “or,” we shade anything that is a solution to at least one of the individual inequalities.
The easiest way to graph a system of two or more linear inequalities is to graph each one individually, using the three steps outlined above.
In this section, we will go over common examples of problems involving linear inequalities and their step-by-step solutions.
Graph the inequality x>2.
Example 1 Solution
First, we need to find the line x=2.
This is the vertical line that is two units to the right of the origin.
Now, we need to decide whether to use a solid or dashed line. Since this inequality uses a greater than sign instead of a greater than or equal to sign, we will use a dashed line.
Finally, this is a vertical line, and we are using a “greater than” sign. Thus, we will shade to the right.
This gives the us the graph below.
Graph the inequality y≤3.
Example 2 Solution
Just like last time, we will find the graph of the line y=3. This is the line that is horizontal and three units above the origin.
Since this graph is a less than or equal to sign instead of just a less than sign, we will use a solid line.
Finally, because this line is less than instead of greater than, we will shade below the line. The result is the graph shown below.
Graph the inequality y≤x. Compare this to the graph of y≥x.
Example 3 Solution
We have two inequalities to graph here, but they use the same line. We need to start by graphing y=x, which is the line that passes through the origin with a slope of 1.
Both inequalities include “equal to,” so both inequalities will have a solid line instead of a dashed line as the bound.
The first line asks us to graph an inequality that is “greater than or equal to.” This means we will shade above the line as shown.
The second inequality has a “less than or equal to” sign, so we must shade below the line.
The only points that these two lines have in common is the line y=x.
Graph the system of inequalities y≥x-1 and y≤2.
Example 4 Solution
We have two lines to graph here. The first is y=x-1. This line has a slope of 1 and the y-intercept (0, -1). The second is y=2, which is a horizontal line that lies two units above the origin.
Both of these lines include the “equal to,” so both of these lines are solid, not dashed.
Now, we need to decide whether to shade above or below the lines. The first line, y=x-1, is greater than, so we will shade above the line. The second inequality is less than, so we will shade below the line.
Since this system is connected by an “and,” we will only shade the overlap of these two inequalities, shown in purple below.
Graph the system of inequalities y≥2x or y≤-2x+1.
Example 5 Solution
Again, we have two inequalities, and we will start by graphing the lines. The line y=2x has a slope of 2 and a y-intercept of 0. The other has a slope of -2 and a y-intercept 1.
Both lines will have solid lines because both include the equality.
The first inequality is greater than or equal to, so we will shade above the solid line. On the other hand, the other inequality is less than or equal to, so will shade below this solid line.
This system of inequalities is connected by a mathematical “or,” so we shade any region that is part of the solution to either inequality, including the overlap.