Linear Inequalities

In this lab you will study mathematical sentences like $y<1$ and $3x+2y<5$, and learn about their solutions. For example, some solutions to $y<1$ are 0.5, 0, −1, and −28. However, not all solutions can be listed. One way to describe all possible solutions is to graph them.

Question 1. All of the blue points on the grid to the left satisfy the equation $y=1$. Name a new point that satisfies $y=1$ (that is, a point with $y$-coordinate equal to 1).

We want to graph the solution set of $y=1$. This means that we want to show all of the points whose $y$-coordinate is 1. Click on to graph the solution set of $y=1$. What is the shape of the solution set?

Question 2. As you saw in Question 1, the graph of the solution set of $y=1$ is a straight line. Now, let's see what the solution set of $y<1$ looks like.

Click on to see the next example. The blue line is the graph of the line $y=1$. All of the blue points on the grid have $y$-coordinate less than 1. Are these points above, below, or on the line $y=1$?

All of the blue points on the grid are in the solution set of $y<1$. Name a new point that satisfies $y<1$ (that is, a point with $y$-coordinate less than 1).

Is it possible to graph the solution set of $y<1$ using a straight line?

How do you think we can graph all of the points that satisfy $y<1$?

The convention we use to graph $y<1$ is to shade the region where all of the points in the solution set lie. So, the region below the line $y=1$ is shaded. Click on to look at the graph of $y<1$. Does the point you found above lie in the shaded region?

What do you think the graph of the solution set of $y>1$ looks like?

Question 3. Click on to look at the graph of $y<mx+b$. Leave $m$ at 0 and use the slider for $b$ to change its value. Notice how changing $b$ affects the solution set of $y<b$.

Using the slider for $b$, find an inequality that has all of the red points on the grid in its solution set. (The inequality is written under the grid.)

Using the slider for $b$, find an inequality that has none of the red points on the grid in its solution set.

Slide $b$ to 0. Which red points are in the solution set of the inequality ($y<0$) now?

Without changing $b$, use the slider for $m$ to change its value and notice how changing $m$ affects the solution set of the inequality $y<mx$. Set $m$ to 1. What is the inequality you are looking at now?

Which of the red points are in the solution set of this inequality?

Name the red points that aren't in the solution set of this inequality.

Question 4. Now we want to see what the solution set of $x>1$ looks like.

Click on . The line to the right of the $y$-axis is the graph of the equation $x=1$. All of the blue points on the grid have $x$-coordinate greater than 1. Are these points to the right, to the left, or on the line $x=1$?

All of the blue points on the grid are in the solution set of $x>1$. Name a new point that satisfies $x>1$ (that is, a point with $x$-coordinate greater than 1).

Is it possible to graph the solution set of $x>1$ using a straight line?

How do you think we can graph all of the points that satisfy $x>1$?

The convention we use to graph $x>1$ is to shade the region where all of the points in the solution set lie. So, the region to the right of the line $x=1$ is shaded. Click on to look at the graph of $x>1$. Does the point you found above lie in the shaded region?

What do you think the graph of the solution set of $x<1$ looks like?

Question 5. Click on . We want to find all of the points that satisfy the inequality $3x+2y<5$. The graph of $3x+2y=5$ is shown on the grid to the left. The solution set of $3x+2y<5$ will be the points on one side or the other of this line. Let's consider the point $(0,0)$ and check if its coordinates satisfy $3x+2y<5$.

Substituting $(x,y) = (0,0)$ into $3x+2y<5$, we have:

$$\table 3(0),+,2(0),<,5; 0,+,0,<,5$$

Is $0<5$? Yes. So the point $(0,0)$ is in the solution set.

Use the method shown above to check the point $(0,4)$. Is $(0,4)$ in the solution set of $3x+2y<5$?

Click on to look at the graph of $3x+2y<5$. Is the point $(0,0)$ in the shaded region?

Is the point $(0,4)$ in the shaded region?

Question 6. So far, you have only looked at inequalities that involved the $>$ and $<$ signs. Now we will look at inequalities that involve the $≥$ and $≤$ signs.

Look back at the graph of $3x+2y<5$. Let's see if the point $(3,-2)$ satisfies this inequality:

$$\table 3(3),+,2(-2),<,5,?; 9,+,-4,<,5,?; ,5,,<,5,?,\text"No"$$

Does the point $(1,1)$ satisfy the inequality?

Does the point $(-1,4)$ satisfy the inequality $3x+2y≤5$?

Click on to see the graph of the solution set of $3x+2y≤5$. Find the points $(3,-2)$, $(1,1)$ and $(-1,4)$ on the graph.

Is $(1,1)$ in the solution set of $3x+2y≤5$?

When an inequality involves the $≥$ or $≤$ signs, we graph its solution set by including a solid line (for example the line $3x+2y=5$ in this case). Why do you think we do so?

Question 7. Click on to see the example for question 7. Use the sliders for $m$ and $b$ to find the solution set shown below. What is the inequality with this solution set?

$y≥\,$

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