Graphing Linear Inequalities Lesson

At this point, we know how to graph a linear equation in two variables. To perform this task quickly, we can solve the equation for y, and place the equation in slope-intercept form:
y = mx + b
This form allows us to observe the y-intercept (0,b) and the slope (m). We can plot the y-intercept as a point on the line and then plot additional points using the slope. When we encounter a linear inequality in two variables, our first objective is to plot what is known as the boundary line. The boundary line separates the solution region from the non-solution region. Every ordered pair in the solution region satisfies the inequality and every ordered pair in the non-solution region does not satisfy the inequality. In order to graph the boundary line, we replace the inequality symbol with an equality symbol and then we graph our line. Since we have different types of inequalities, the boundary line will change based on the given scenario:
"<" or ">" » dashed line
"≤" or "≥" » solid line
If we see a strict inequality, we don't allow for the boundary line to be part of the solution. This will be the case for strictly less than "<" or strictly greater than ">". For this scenario, we graph our boundary line as a dashed or broken line. This indicates that the line is not part of the solution. If we see a non-strict inequality, we do allow for the boundary line to be part of the solution. This will be the case with less than or equal to "≤" or greater than or equal to "≥". For this scenario, we graph our boundary line as a solid line. This indicates that points on the line will satisfy the inequality.
Let's suppose we saw the following linear inequality in two variables:
2x + 5y > 10
Let's first solve the inequality for y:
5y > -2x + 10
5/5 y > -2/5 x + 10/5
y > -2/5 x + 2
We can replace the inequality with an equality:
y = -2/5 x + 2
This will be our boundary line. We need to note that this is a strict inequality. The boundary line will be a dashed line. To graph this boundary line, we can plot the point (0,2), our y-intercept: Now we can use our slope (-2/5) to obtain additional points. We can rise (-2) and run (5) to obtain the point: (5,0). We could also reverse this and write our slope as (2/-5). We can start at (0,2) and rise (2) and run (-5) to obtain the point: (-5,4): Let's graph our boundary line by drawing a dashed line through our three points: Now that our boundary is drawn, we want to shade the solution region. We have two methods to perform this action. If the inequality is solved for y, we can use the following rule:
> or ≥ » shade above the line
< or ≤ » shade below the line
In other words, shade above the line for a greater than or greater than or equal to and shade below the line for a less than or a less than or equal to. Since our inequality when solved for y was a greater than (y > -2/5 x + 2), we want to shade above the line: Additionally, we can use a test point. The test point method is a bit more tedious, but you may see this covered in your class. We know points on the coordinate plane fall in one of three categories:
1) Lie in the solution region
2) Lie in the non-solution region
3) Lie on the boundary line
We choose a point that is not on the boundary line and substitute into our inequality. If we obtain a true statement, that point lies in the solution region. We can then shade the side of the boundary where the point lies. On the other hand, we may see that our point does not satisfy the inequality. This tells us that our point lies in the non-solution region. For this situation, we will want to shade the other side of the boundary line.
For our example, we can choose the point (0,0) as it is very easy to work with and it is not on the boundary line. Let's plug into our inequality:
2x + 5y > 10
2(0) + 5(0) > 10
0 > 10 (false)
Since our point did not satisfy the inequality, we know (0,0) lies in the non-solution region. This tells us to shade the other region, which means we shade above the line: We can see we obtain the same result either way. Solving for y and shading based on the symbol is much faster than using a test point. Let's look at a few examples.
Example 1: Graph each Linear Inequality in Two Variables.
x - 5y ≤ 10
Let's solve the inequality for y:
-5y ≤ -x + 10
-5/-5 y ≥ -1/-5 x + 10/-5
y ≥ 1/5 x - 2
We can graph our solid boundary line as:
y = 1/5 x - 2 We see the boundary line is solid since we had a non-strict inequality. Now we can shade above the line since we had a greater than or equal to: Example 2: Graph each Linear Inequality in Two Variables.
2x - 3y > 6
Let's solve the inequality for y:
-3y > -2x + 6
-3/-3 y < -2/-3 x + 6/-3
y < 2/3 x - 2
We can graph our dashed boundary line as:
y = 2/3 x - 2 We see the boundary line is solid since we have a strict inequality. Now we can shade below the line since we had a less than: Example 3: Graph each Linear Inequality in Two Variables.
x ≤ -3
This is a special case scenario. When we see the format of: x = some number, we have a vertical line. We graph the solid boundary line as:
x = -3 In this case, we have to think about the solution region. Since x can be any value that is less than or equal to (-3), we want to shade to the left of our boundary line. We are shading to the left of the boundary line since values on the x-axis decrease as we move left:

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