Chapter 4.4

Linear Inequalities in Two Variables

In chapter 3 we learned how to graph Linear equations in two variables. In this section we will graph linear inequalities in two variables. I will show you how to do the them by hand and then with the calculator.

Example 1. Graph by hand.

Step 1. Graph as you would an equality.

Click image to enlarge

We now need to choose a test point that is not on the line. The point will always work unless your line goes through the origin. Substitute your test point into the inequality to see if it satisfies it. If it does, then shade toward your test point; otherwise, shade away from it.

Because it is true, we are going to shade toward the test point (up, in this case).

Your graph should look similar to the one above.

Let’s graph the same inequality with the calculator:

	Step 1. Hit he [Y=] key and graph it just like you would and equality.
	Step 2. In order to make it shade, you need to scroll over to the far left…until you see the diagonal line flashing. Then hit [ENTER] key until you see the triangle shaded above (because ).
	Step 3. Let’s get the standard viewing window. Hit the [ZOOM] key and scroll to ZStandard. This gives you a 10 by 10 viewing window.
	Step 4. You should get the screen on the left.

Example 2. Graph by hand.

Step 1. It is probably easiest to find the x and y intercepts and draw a line through them. When we plug in a 0 for x we get for y and when we plug in a 0 for y, we get a 4 for x. This is summarized with a table below.

x	y
0
	0

Plot these points and draw a line through them:

Selecting a test point of , we get:

Thus, our final shading looks like this:

Again, lets show how to use the TI-83.

Graph with the TI-83:

Our first step must be done manually:

Step 1. Solve for y.

Subtract from both sides:

Divide by

	Step 1. We must first solve for y (see above). This may take a bit of effort. Note that because we divided by a negative number, we must flip the inequality sign. Also, it is important that you place parentheses around the numerator.
	Step 2. Scroll over to change the diagonal line to an upper triangle as shown on the left (hit the [ENTER] key twice).
	Step 3. Hit the [ZOOM] and Zstandard to graph. This puts in the standard viewing window.
	Step 4. We could set the window similar to the one I did by hand.
	Step 5. This looks a bit more line the one I did by hand. You did not have to do steps 4 or 5.

Graphing Horizontal and Vertical lines

Example 3. Graph the line .

Step 1. Graph the vertical line first, then shade in the direction of the inequality.

Example 4. Graph the inequality .

Note the dashed line being used because of the strict inequality (not equal to)

System of Inequalities

Example 5. Graph the system of inequalities by hand.

Just graph each inequality separately on the same x-y axes.

Note that a dashed line is used whenever we have a strict inequality ().

Plugging the test point of in for both equations produces false results. Therefore, we should shade away from the test point in both cases. When doing this by hand, the easiest way to do this is by using indication arrows.

We are looking for where the shadings intersect. In this example, this occurs in the far right region. So we make a final shading as follows

We will do the same example on the TI-83:

Example 6. Graph the system of inequalities by using the TI-83.

Again we must first solve for y in both equations. I will leave this up to you.

	Step 1. Plug in equations to Y1 and Y2
	Step 2. Now we must change the diagonal line to the appropriate triangle (above or below). Note: Because we divided by a , we must flip the inequality sign.
	Step 3. Hit the [ZOOM] Zstandard key. This looks like the one we did by hand.

Example 7. Graph and find any vertices formed:

We will look at the graph drawn by hand first. Just graph the first two equations as we did the previous example. Then the last two constraints () just use indication arrows. This constraint is used quite often in application problems because x and y will represent something and many times that something cannot be negative (e.g. x=the number of chairs and y=the number of tables).

Note where the shadings come together. Make your final shading.

Now let’s view it on the Grapher:

	Step 1. Solve the first two equations for y and plug them into Y1 and Y2 respectively and determine the direction of the shading. Your screen should look like the one on the left
	Step 2. Instead of trying to graph the last two constraints, let’s just adjust our window. If you try to graph the last two you will have trouble determining where the final shading occurs. We will adjust the Xmin to 0 and the Ymin to 0.
	Step 3. Hit the [GRAPH] key.
	Step 4. If you want to get a better look at the graph, change your Xmax and Ymax to 6.
	Step 5. We now need to find the Points along the feasible region (the region where the shadings intersect). You will need to use the intersect feature we did in section 3.3.. Hit [2^nd] [CALC] [intersect] then hit [ENTER] 3 times. For the details go to Chapter 3.3.
	Step 6. So the vertices formed are as follows: The y-intercept of the feasible region. The x-intercept of the feasible region. The origin—which is on the feasible region. The intersection of the two inequalities.