Graph Linear Inequalities in Two Variables

3.4 Graph Linear Inequalities in Two Variables

Topics covered in this section are:

  1. Verify solutions to an inequality in two variables.
  2. Recognize the relation between the solutions of an inequality and its graph.
  3. Graph linear inequalities in two variables.
  4. Solve applications using linear inequalities in two variables.

3.4.1 Verify Solutions to an Inequality in Two Variables

Previously we learned to solve inequalities with only one variable. We will now learn about inequalities containing two variables. In particular we will look at linear inequalities in two variables which are very similar to linear equations in two variables.

Linear inequalities in two variables have many applications. If you ran a business, for example, you would want your revenue to be greater than your costs—so that your business made a profit.

LINEAR INEQUALITY

linear inequality is an inequality that can be written in one of the following forms:
$\ \ \ \ \ \ \ Ax+By > C \ \ \ \ \ \ \ Ax+By≥C \ \ \ \ \ \ \ Ax+By<C \ \ \ \ \ \ \ Ax+By≤C \ \ \ \ \ \ \ \ $
Where $A$ and $B$ are not both zero.

Recall that an inequality with one variable had many solutions. For example, the solution to the inequality $x>3$ is any number greater than $3$. We showed this on the number line by shading in the number line to the right of $3$, and putting an open parenthesis at $3$. See Figure 3.10.

Image of the number line with the integers from negative 5 to 5. The part of the number line to the right of 3 is marked with a blue line. The number 3 is marked with a blue open parenthesis.
Figure 3.10

Similarly, linear inequalities in two variables have many solutions. Any ordered pair $(x, y)$ that makes an inequality true when we substitute in the values is a solution to a linear inequality.

SOLUTION TO A LINEAR INEQUALITY

An ordered pair $(x, y)$ is a solution to a linear inequality if the inequality is true when we substitute the values of $x$ and $y$.

Example 1

Determine whether each ordered pair is a solution to the inequality $y>x+4$:

  • $(0, 0)$
  • $(1, 6)$
  • $(2, 6)$
  • $(-5, -15)$
  • $(-8, 12)$
Solution

Part 1

$(0, 0)$$y>x+4$
Subsitute $\textcolor{red}{0}$ for $x$ and $\textcolor{blue}{0}$ for $y$.$\textcolor{blue}{0} > \textcolor{red}{0} +4$
Simplify.$0 \cancel{>} 4$
So, $(0, 0)$ is not a solution to $y>x+4$.

Part 2

$(1, 6)$$y>x+4$
Subsitute $\textcolor{red}{1}$ for $x$ and $\textcolor{blue}{6}$ for $y$.$\textcolor{blue}{6} > \textcolor{red}{1} +4$
Simplify.$6 > 5$
So, $(1, 6)$ is a solution to $y>x+4$.

Part 3

$(2, 6)$$y>x+4$
Subsitute $\textcolor{red}{2}$ for $x$ and $\textcolor{blue}{6}$ for $y$.$\textcolor{blue}{6} > \textcolor{red}{2} +4$
Simplify.$6 \cancel{>} 6$
So, $(2, 6)$ is not a solution to $y>x+4$.

Part 4

$(-5, -15)$$y>x+4$
Subsitute $\textcolor{red}{-5}$ for $x$ and $\textcolor{blue}{-15}$ for $y$.$\textcolor{blue}{-15} > \textcolor{red}{-5} +4$
Simplify.$-15 \cancel{>} -1$
So, $(-5, -15)$ is not a solution to $y>x+4$.

Part 5

$(-8, 12)$$y>x+4$
Subsitute $\textcolor{red}{-8}$ for $x$ and $\textcolor{blue}{12}$ for $y$.$\textcolor{blue}{12} > \textcolor{red}{-8} +4$
Simplify.$12 > -4$
So, $(-8, 12)$ is a solution to $y>x+4$.

3.4.2 Recognize the Relation Between the Solutions of an Inequality and its Graph

Now, we will look at how the solutions of an inequality relate to its graph.

Let’s think about the number line in shown previously again. The point $x=3$ separated that number line into two parts. On one side of $3$ are all the numbers less than $3$. On the other side of $3$ all the numbers are greater than $3$. See Figure 3.11.

Image of the number line with the integers from negative 5 to 5. The part of the number line to the right of 3 is marked with a blue line. The number 3 is marked with a blue open parenthesis. The part of the number line to the right of 3 is labeled “numbers greater than 3”. The part of the number line to the left of 3 is labeled “numbers less than 3”.
Figure 3.11 The solution to $x>3$ is the shaded part of the number line to the right of $x=3$.

Similarly, the line $y=x+4$ separates the plane into two regions. On one side of the line are points with $y<x+4$. On the other side of the line are the points with $y>x+4$. We call the line $y=x+4$ a boundary line.

BOUNDARY LINE

The line with equation $Ax+By=C$ is the boundary line that separates the region where $Ax+By>C$ from the region where $Ax+By<C$.

For an inequality in one variable, the endpoint is shown with a parenthesis or a bracket depending on whether or not $a$ is included in the solution:

Two number lines are shown with the middle labeled with the number “a”. In both number lines, the part to the left of the number a is marked with red. The first number line is labeled “x is less than a” and the number a is marked with an open parenthesis. The second number line is labeled “x is less than or equal to a” and the number a is marked with an open bracket.

Similarly, for an inequality in two variables, the boundary line is shown with a solid or dashed line to show whether or not it the line is included in the solution.

$Ax+By<C$$Ax+By≤C$
$Ax+By>C$$Ax+By≥C$
Boundary line is $Ax+By=C$Boundary line is $Ax+By=C$
Boundary line is not included in solution.Boundary line is included in solution.
Boundary line is dashed.Boundary line is solid.

Now, let’s take a look at what we found in Example 1. We’ll start by graphing the line $y=x+4$, and then we’ll plot the five points we tested, as shown in the graph. See Figure 3.12.

This figure has the graph of some points and a straight line on the x y-coordinate plane. The x and y axes run from negative 16 to 16. The points (negative 8, 12), (negative 5, negative 15), (0, 0), (1, 6), and (2, 6) are plotted and labeled with their coordinates. A straight line is drawn through the points (negative 4, 0), (0, 4), and (2, 6)."
Figure 3.12

In Example 1 we found that some of the points were solutions to the inequality $y>x+4$ and some were not.

Which of the points we plotted are solutions to the inequality $y>x+4$?

The points $(1, 6)$ and $(−8, 12)$ are solutions to the inequality $y>x+4$. Notice that they are both on the same side of the boundary line $y=x+4$.

The two points $(0, 0)$ and $(−5, −15)$ are on the other side of the boundary line $y=x+4$, and they are not solutions to the inequality $y>x+4.$ For those two points, $y<x+4$.

What about the point $(2, 6)$? Because $6=2+4$, the point is a solution to the equation $y=x+4$, but not a solution to the inequality $y>x+4$. So the point $(2,6)$ is on the boundary line.

Let’s take another point above the boundary line and test whether or not it is a solution to the inequality $y>x+4$. The point $(0, 10)$ clearly looks to above the boundary line, doesn’t it? Is it a solution to the inequality?

$\begin{align*} y&>x+4 \\ 10&>0+4 \\ 10&>4 \end{align*}$

So, $(0, 10)$ is a solution to $y>x+4$.

Any point you choose above the boundary line is a solution to the inequality $y>x+4$. All points above the boundary line are solutions.

Similarly, all points below the boundary line, the side with $(0, 0)$ and $(−5, −15)$, are not solutions to $y>x+4$, as shown in Figure 3.13.

This figure has the graph of some points and a straight line on the x y-coordinate plane. The x and y axes run from negative 16 to 16. The points (negative 8, 12), (negative 5, negative 15), (0, 0), (1, 6), and (2, 6) are plotted and labeled with their coordinates. A straight line is drawn through the points (negative 4, 0), (0, 4), and (2, 6). The line divides the x y-coordinate plane into two halves. The top left half is labeled y is greater than x plus 4. The bottom right half is labeled y is less than x plus 4.
Figure 3.13

The graph of the inequality $y>x+4$ is shown in below.

The line $y=x+4$ divides the plane into two regions. The shaded side shows the solutions to the inequality $y>x+4$.

The points on the boundary line, those where $y=x+4$, are not solutions to the inequality $y>x+4$, so the line itself is not part of the solution. We show that by making the line dashed, not solid.

This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 8 to 8. A straight dashed line is drawn through the points (negative 4, 0), (0, 4), and (2, 6). The line divides the x y-coordinate plane into two halves. The top left half is colored red to indicate that this is where the solutions of the inequality are.

Example 2

The boundary line shown in this graph is $y=2x−1$. Write the inequality shown by the graph.

This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 8 to 8. A straight dashed line is drawn through the points (0, negative 1), (1, 1), and (2, 3). The line divides the x y-coordinate plane into two halves. The top left half is colored red to indicate that this is where the solutions of the inequality are.
Solution

The line $y=2x−1$ is the boundary line. On one side of the line are the points with $y>2x−1$ and on the other side of the line are the points with $y<2x−1$.

Let’s test the point $(0, 0)$ and see which inequality describes its position relative to the boundary line.

At $(0, 0)$, which inequality is true: $y>2x-1$ or $y<2x-1$?

$y>2x-1$$y<2x-1$
$0>2 \cdot 0-1$$0<2 \cdot 0 – 1$
$0 > -1$ True$0 < -1$ False

Since, $y>2x−1$ is true, the side of the line with $(0, 0)$, is the solution. The shaded region shows the solution of the inequality $y>2x−1$.

Since the boundary line is graphed with a solid line, the inequality includes the equal sign.

The graph shows the inequality $y≥2x−1$.

We could use any point as a test point, provided it is not on the line. Why did we choose $(0, 0)$? Because it’s the easiest to evaluate. You may want to pick a point on the other side of the boundary line and check that $y<2x−1$.

Example 3

The boundary line shown in this graph is $2x+3y=6$. Write the inequality shown by the graph.

This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 8 to 8. A straight dashed line is drawn through the points (0, 2), (3, 0), and (6, negative 2). The line divides the x y-coordinate plane into two halves. The bottom left half is colored red to indicate that this is where the solutions of the inequality are.
Solution

The line $2x+3y=6$ is the boundary line. On one side of the line are the points with $2x+3y>6$ and on the other side of the line are the points with $2x+3y<6$.

Let’s test the point $(0, 0)$ and see which inequality describes its side of the boundary line.

At $(0, 0)$, which inequality is true: $2x+3y>6$ or $2x+3y<6$?

$2x+3y>6$$2x+3y<6$
$2(0)+3(0)>6$$2(0)+3(0)<6$
$0>6$ False$0 < 6$ True

So the side with $(0, 0)$ is the side where $2x+3y<6$.

(You may want to pick a point on the other side of the boundary line and check that $2x+3y>6$.)

Since the boundary line is graphed as a dashed line, the inequality does not include an equal sign.

The shaded region shows the solution to the inequality $2x+3y<6$.

3.4.3 Graph Linear Inequalities in Two Variables

Now that we know what the graph of a linear inequality looks like and how it relates to a boundary equation we can use this knowledge to graph a given linear inequality.

Example 4

Graph the linear inequality $y≥\frac{3}{4}x-2$.

Solution
Step 1 is to Identify and graph the boundary line. If the inequality is less than or equal or greater than or equal, the boundary line is solid. If the inequality is less than or greater than, the boundary line is dashed. In this example the inequality sign is greater than or equal, so we draw a solid line. Replace the inequality sign with an equal sign to find the boundary line. Graph the boundary line y = 3 divided by 4 times x minus 2. The figure then shows the graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The line goes through the points (0, negative 2), (4, 1), and (8, 4).
Step 2 is to test a point that is not on the boundary line. Is it a solution of the inequality? We will test (0, 0). At (0, 0) is y greater than or equal to 3 divided by 4 times x minus 2? Is 0 greater than or equal to 3 divided by 4 times 0 minus 2? 0 is greater than or equal to negative 2 so (0, 0) is a solution.
Step 3 is to shade in one side of the boundary line. If the test point is a solution, shade in the side that includes the point. If the test point is not a solution, shade in the opposite side. The test point (0, 0), is a solution to y greater than or equal to 3 divided by 4 times x minus 2. So we shade in the side that contains (0, 0). The figure then shows the graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 12 to 12. The line goes through the points (0, negative 2), (4, 1), and (8, 4). The top left half of the coordinate plane is shaded to indicate that this is where the solution set is located.

The steps we take to graph a linear inequality are summarized here.

HOW TO: Graph a linear inequality in two variables.

  1. Identify and graph the boundary line.
    • If the inequality is $≤$ or $≥$, the boundary line is solid.
    • If the inequality is $<$ or $>$, the boundary line is dashed.
  2. Test a point that is not on the boundary line. Is it a solution of the inequality?
  3. Shade in one side of the boundary line.
    • If the test point is a solution, shade in the side that includes the point.
    • If the test point is not a solution, shade in the opposite side.

Example 5

Graph the linear inequality $x-2y<5$.

Solution

First, we graph the boundary line $x-2y=5$. The inequality is $<$ so we draw a dotted line.

This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 8 to 8. A straight dashed line is drawn through the points (negative 3, negative 4), (1, negative 2), and (5, 0).

Then, we test a point. We’ll use $(0, 0)$ again because it is easy to evaluate and it is not on the boundary line.

Is $(0, 0)$ a solution of $x-2y<5$?

$\begin{align*} \textcolor{blue}{0} -2(\textcolor{blue}{0}) &< 5 \\ 0 – 0 &<5 \\ 0&<5 \end{align*}$

The point $(0, 0)$ is a solution of $x-2y<5$, so we shade in that side of the boundary line.

This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 8 to 8. A straight dashed line is drawn through the points (negative 3, negative 4), (1, negative 2), and (5, 0). The line divides the x y-coordinate plane into two halves. The top left half is shaded red to indicate that this is where the solutions of the inequality are.

All the points in the shaded region, but not those on the boundary line, represent the solution to $x-2y<5$.

What if the boundary line goes through the origin? Then, we won’t be able to use $(0, 0)$ as a test point. No problem—we’ll just choose some other point that is not on the boundary line.

Example 6

Graph the linear inequality: $y≤-4x$.

Solution

First, we graph the boundary line $y=-4x$. It is in slope-intercept form, with $m=-4$ and $b=0$. The inequality is $≤$ so we draw a solid line.

This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 8 to 8. A straight is drawn through the points (0, 0), (1, negative 4), and (negative 1, 4).

Now we need a test point. We can see that the point $(1, 0)$ is not on the boundary line.

Is $(1, 0)$ a solution of $y≤-4x$?

$\textcolor{blue}{0} ≤-4(1)$
$0 \cancel{≤} 4$

The point $(1, 0)$ is not a solution to $y≤-4x$, so we shade the opposite side of the boundary line.

The image shows the graph of the solid boundary line y= -4x with the shaded region to the left of the boundary line. The test point (1,0) was not a solution to the inequality y≤-4x.

All points in the shaded region and on the boundary line represent the solutions to $y≤-4x$.

Some linear inequalities have only one variable. They may have an $x$ but no $y$, or a $y$ but no $x$. In these cases, the boundary line will be either a vertical or a horizontal line.

Recall that:

$x=a$vertical line
$y=b$horizontal line

Example 7

Graph the linear inequality: $y>3$.

Solution

First, we graph the boundary line $y=3$. It is a horizontal line. The inequality is $>$ so we draw a dashed line.

We test the point $(0, 0)$.

$y>3$
$0\cancel{>}3$

So, $(0, 0)$ is not a solution to $y>3$.

We shade the side the does not include $(0, 0)$ as shown in this graph.

This figure has the graph of a straight horizontal dashed line on the x y-coordinate plane. The x and y axes run from negative 8 to 8. A horizontal dashed line is drawn through the points (negative 1, 3), (0, 3), and (1, 3). The line divides the x y-coordinate plane into two halves. The top half is shaded red to indicate that this is where the solutions of the inequality are.

All points in the shaded region, but not those on the boundary line, represent the solutions to $y>3$.

3.4.4 Solve Applications using Linear Inequalities in Two Variables

Many fields use linear inequalities to model a problem. While our examples may be about simple situations, they give us an opportunity to build our skills and to get a feel for how they might be used.

Example 8

Hilaria works two part time jobs in order to earn enough money to meet her obligations of at least $\$240$ a week. Her job in food service pays $\$10$ an hour and her tutoring job on campus pays $\$15$ an hour. How many hours does Hilaria need to work at each job to earn at least $\$240$?

  • Let $x$ be the number of hours she works at the job in food service and let $y$ be the number of hours she works tutoring. Write an inequality that would model this situation.
  • Graph the inequality.
  • Find three ordered pairs $(x, y)$ that would be solutions to the inequality. Then, explain what that means for Hilaria.
Solution

Part 1

We let $x$ be the number of hours she works at the job in food service and let $y$ be the number of hours she works tutoring.

She earns $\$10$ per hour at the job in food service and $\$15$ an hour tutoring. At each job, the number of hours multiplied by the hourly wage will gives the amount earned at that job.

10 x plus 15 y is greater than 240. The “10 x” is labeled “Amount earned at the food service job”. The “15 y” is labeled “the amount earned tutoring”. The “is greater than 240” is labeled “is at least 240”.

Part 2

To graph the inequality, we put it in slope-intercept form.

$\begin{align*} 10x+15y&≥240 \\ 15y&≥-10x+240 \\ y&≥-\frac{2}{3}x+16 \end{align*}$

This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from 0 to 25. A line is drawn through the points (0, 16), (15, 6), and (24, 0). The line divides the x y-coordinate plane into two halves. The line and the top right half are shaded red to indicate that this is where the solutions of the inequality are.

Part 3

From the graph, we see that the ordered pairs $(15, 10)$, $(0, 16)$, and $(24, 0)$ represent three of infinitely many solutions. Check the values in the inequality.

$(15, 10)$$(0, 16)$$(24, 0)$
$10x+15y≥240$$10x+15y≥240$$10x+15y≥240$
$10(\textcolor{red}{15})+ 15(\textcolor{red}{10})≥240$$10(\textcolor{red}{0}) + 15(\textcolor{red}{16})≥240$$10(\textcolor{red}{24}) + 15(\textcolor{red}{0})≥240$
$300≥240$ True$240≥240$ True$240≥240$ True

For Hilaria, it means that to earn at least $\$240$, she can work $15$ hours tutoring and $10$ hours at her fast-food job, earn all her money tutoring for $16$ hours, or earn all her money while working $24$ hours at the job in food service.

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