Section 8.4: Linear Inequalities In Two Variables
Because we are dealing with two variables now, this means we need to graph on the x-y plane. This is similar to
graphing lines except as you can see we now deal with inequalities. Make sure you can graph lines (either by
plotting points OR by using the slope and y-intercept). The solutions to linear inequalities, however, is a little
different than what we have seen before.
§1 Linear Inequalities
Let’s take a look at an example first. Say we want to graph the solution of 2 x − y < 6 . Notice that there is no
equal sign here. So how does this change the problem?
Because we are dealing with linear inequalities, we need to understand what the problem is asking. We need to
find all the (x,y) ordered pairs that are solutions to the inequality. Remember, the points that lie on the line
2x − y =
6 are the solutions to the linear equation. For example, the points (3,0), (4,2) and (0,-6) are some of
the many solutions to the linear equation.
But we are dealing with inequalities now. It turns out that the graph of the line in question is simply a boundary
line. The line divides the x-y plane into two half-planes. Each half-plane represents a set of points. It’s your job to
determine which half-plane contains the points that are the solution.
The best thing to do is pick a test point not on the line and plug it into the inequality. If the point works, or
satisfies the original inequality, then the half-plane that the point is on contains all the solutions, so we shade
that side.
If the test point does not work, or does not satisfy the original inequality, then the half-plane that the point is in
does not contain any solution. Hence you must shade the other side of the boundary. This shaded region is also
called the feasible region, because it contains all the points that do work!
For example, the graph of 2 x − y =
6 above looks like the following:
But we are actually trying to solve the inequality 2 x − y < 6 . Since this is strictly less than, we would actually
have to use a dotted line instead of a solid line. The dotted line means that the points on the line ARE NOT
solutions to the inequality.
Next, we pick a test point not on the line. The easiest test point to pick is (0,0). So plug in 0 for x and 0 for y. We
end up with 0 – 0 < 6, or 0 < 6. This is true! Hence, it means that (0,0) is a solution to the inequality. It also
means that every other point on the same side of the boundary line is also a solution! Hence we shade that side
of the boundary.
So remember, first draw the boundary line. You can plot two points or use the slope and the y-intercept to draw
the boundary line. If the inequality is ≤ or ≥ the boundary line is solid. If the inequality is < or > then the
boundary line is dotted.
Next, choose a test point not on the line and plug it in. The easiest test point to pick is (0,0). Plug in 0 for x and 0
for y and see the result. If the point works, shade the side that the test point is on. However, if the test point
doesn’t work, then shade the other side of the boundary.
Shading represents the points that are in the feasible region. These are all the points that satisfies the inequality.
You must be very careful when graphing these. Sometimes the line may be a vertical or horizontal line, but don’t
get confused by these. Just make sure you understand what the inequality means and how to use a test point to
find the feasible region! If the line passes through the origin, you must use a test point other than (0,0).
PRACTICE
1) Graph 3 x + 4 y ≤ 12
2) Graph x ≥ 3 y
3) Graph x > 4
§2 The Intersection Of Two Linear Inequalities
In these examples, we graph two linear inequalities IN THE SAME PLANE. The intersection then is simply the
region that satisfies both inequalities. If you use a different colored marker for the two inequalities, then the
answer is the region where both shaded parts overlap.
If there is no overlap, then that means that there is no solution. Remember, the solution are the set of points
that works for both inequalities!
For example, say we want to solve x − y ≤ 4 and x ≥ −2
The graph in red represents the first inequality. The graph in blue represents the second inequality.
If you graph them on the same plane, we get the following:
Note that region that is shaded twice. That is the feasible region – the set of points that satisfy both inequalities.
Note the region in white. This represents the set of points that do not work for either inequality.
PRACTICE
1) Graph and denote the feasible region of x + y ≥ 1 and y < 2
2) Graph and denote the feasible region of 2 x − y ≥ 6 or x ≤ −0