Systems of Linear Inequalities

A system of inequalities is a set of two or more inequalities using the same variables. For example the pair of inequalities shown to the right is a system of linear inequalities. In this lab you will learn about solutions to systems of linear inequalities and how to find them by graphing.

$$\{\,\cl"ma-join1-align"{\table x+2y<-7; 2x-3y>0}$$

Question 1. We want to solve this system of linear inequalities:

$$\{\,\cl"ma-join1-align"{\table y≥-2x-2; y≥-2}$$

That is, we want to find the $(x,y)$ pairs that are solutions to both inequalities. First, we need to graph the lines $y=-2x-2$ and $y=-2$. Look at the grids to the left. On the left grid, you are looking at the graph of $y=-2x-2$. On the right, you are looking at the graph of $y=-2$.

Here is a way to see if the point $(1,2)$ is a solution to the first inequality:

$y≥-2x-2$ for $(1,2)$ ?
$2≥-2(1)-2$ ?
$2≥-2-2$ ?
$2≥-4$?
YES

Test each point in the table below in both inequalities. Also indicate which points are solutions to both.

Point	Solution to $y≥-2x-2$?	Solution to $y≥-2$?	Solution to both?
$(1,2)$	yes
$(0,0)$
$(-2,-3)$
$(-3,1)$

Click on to look at the graphs of the inequalities. Are the blue points that are solutions to $y≥-2x-2$ in the shaded region on the left grid?

Are the blue points that are solutions to $y≥-2$ in the shaded region on the right?

Name the blue points on the grid that are solutions to the entire system (both inequalities).

Click on to see $y≥-2x-2$ and $y≥-2$ graphed together. The dark blue section is the area where the solution sets overlap. The light blue section is the rest of the solution set for $y≥-2$ and the pink section is the rest of the solution set for $y≥-2x-2$. Find the points you found in the last question on the graph. Which section (what color) are they in?

What do you think the dark blue area represents about the entire system of inequalities?

Question 2. Now, let's solve the system of linear inequalities:

$$\{\,\cl"ma-join1-align"{\table y≥-4x-2; 3x+4y≤18}$$

First, we need to graph the lines $y=-4x-2$ and $3x+4y=18$. To do so, click on . The red line is the graph of $y=-4x-2$ and the blue line is the graph of $3x+4y=18$.

For each point in the table below, locate the point on the grid and complete the table.

Point	Solution to $y≥-4x-2$?	Solution to $3x+4y≤18$?	Solution to both?
$(-2,0)$
$(4,-4)$
$(6,4)$

Click on to look at the graph of the inequalities. Remember that the dark blue region is the graph of the solution set of the system of linear inequalities. Name the point on the grid that lies in the solution set.

Does your answer match your answer from the table?

Question 3. Now it's your turn to solve the system of linear inequalities shown here:

$$\{\,\cl"ma-join1-align"{\table x+y≥-2; 2x-3y≤6}$$

The graphs of the lines $x+y=-2$ and $2x-3y=6$ are given on the grid below. As you can see, the graphs of these two lines divide the grid into four sections, which have been labeled A, B, C, and D.

Choose a point in each section of the grid above (A, B, C, and D), and fill in the table below.

Section	Point	Solution to $x+y≥-2$?	Solution to $2x-3y≤6$?	Solution to both?
A
B
C
D

Which section(s) (A, B, C, or D) are solutions to $x+y≥-2$?

Which section(s) are solutions to $2x-3y≤6$?

Which section(s) are the solution set to the system of linear inequalities?

Click on to check your answers.

Question 4. Click on to see the example for question 4. Use the sliders to find the system of two inequalities whose solution is the lower triangle shown here. (The inequalities are shown below the graph.)

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