4.7 Graphing Systems of Linear Inequalities

Learning objectives

Learning objectives

• Determine whether an ordered pair is a solution of a system of linear inequalities

• Solve a system of linear inequalities by graphing

• Solve applications of systems of inequalities

Determine whether an ordered pair is a solution of a system of linear inequalities

The definition of a system of linear inequalities is very similar to the definition of a system of linear equations.

A system of linear inequalities looks like a system of linear equations, but it has inequalities instead of equations. A system of two linear inequalities is shown here.

To solve a system of linear inequalities, we will find values of the variables that are solutions to both inequalities. We solve the system by using the graphs of each inequality and show the solution as a graph. We will find the region on the plane that contains all ordered pairs $(x,y)$ that make both inequalities true.

To determine if an ordered pair is a solution to a system of two inequalities, we substitute the values of the variables into each inequality. If the ordered pair makes both inequalities true, it is a solution to the system.

Example 4.53

Solution

ⓐ Is the ordered pair $(-2,4)$ a solution?

The ordered pair $(-2,4)$ made both inequalities true. Therefore $(-2,4)$ is a solution to this system.

ⓑ Is the ordered pair $(3,1)$ a solution?

The ordered pair $(3,1)$ made one inequality true, but the other one false. Therefore $(3,1)$ is not a solution to this system.

Solve a System of Linear Inequalities by Graphing

The solution to a single linear inequality is the region on one side of the boundary line that contains all the points that make the inequality true. The solution to a system of two linear inequalities is a region that contains the solutions to both inequalities. To find this region, we will graph each inequality separately and then locate the region where they are both true. The solution is always shown as a graph.

Example 4.54

Example 4.55

Solution

The point of intersection of the two lines is not included as both boundary lines were dashed. The solution is the area shaded twice—which appears as the darkest shaded region.

Example 4.56

Solution

The point $(0,0)$ is in the solution and we have already found it to be a solution of each inequality. The point of intersection of the two lines is not included as both boundary lines were dashed.

The solution is the area shaded twice—which appears as the darkest shaded region.

Systems of linear inequalities where the boundary lines are parallel might have no solution. We’ll see this in the next example.

Example 4.57

Solution

There is no point in both shaded regions, so the system has no solution.

Some systems of linear inequalities where the boundary lines are parallel will have a solution. We’ll see this in the next example.

Example 4.58

Solution

No point on the boundary lines is included in the solution as both lines are dashed.

The solution is the region that is shaded twice which is also the solution to $x-2y<-4.$

Solve Applications of Systems of Inequalities

The first thing we’ll need to do to solve applications of systems of inequalities is to translate each condition into an inequality. Then we graph the system, as we did above, to see the region that contains the solutions. Many situations will be realistic only if both variables are positive, so we add inequalities to the system as additional requirements.

Example 4.59

Solution

ⓐ

• Let $x=\mathrm{the number of small photos}.$

• $\,y=\mathrm{the number of large photos}$

• To find the system of equations translate the information.

• $\,\mathrm{She wants to have at least 25 photos.}$

• $\,\mathrm{The number of small plus the number of large should be at least 25.}$

• $x+y\ge{}25$

• $\,\text{\textdollar 4 for each small and \$10 for each large must be no more than \$200}$

• $4x+10y\le{}200$

• $\,\mathrm{The number of small photos must be greater than or equal to 0.}$

• $x\ge{}0$

• $\,\mathrm{The number of large photos must be greater than or equal to 0.}$

• $y\ge{}0$

• We have our system of equations.

• $\,\mathrm{\{}\begin{array}{l} x+y\ge{}25 \\ 4x+10y\le{}200 \\ x\ge{}0 \\ y\ge{}0 \\ \end{array}\,$

ⓑ Since $x\ge{}0$ and $y\ge{}0$ (both are greater than or equal to) all solutions will be in the first quadrant. As a result, our graph shows only quadrant one.

The solution of the system is the region of the graph that is shaded the darkest. The boundary line sections that border the darkly-shaded section are included in the solution as are the points on the x-axis from (25, 0) to (55, 0).

ⓒ To determine if 10 small and 20 large photos would work, we look at the graph to see if the point (10, 20) is in the solution region. We could also test the point to see if it is a solution of both equations.

It is not, Christy would not display 10 small and 20 large photos.

ⓓ To determine if 20 small and 10 large photos would work, we look at the graph to see if the point (20, 10) is in the solution region. We could also test the point to see if it is a solution of both equations.

It is, so Christy could choose to display 20 small and 10 large photos.

Notice that we could also test the possible solutions by substituting the values into each inequality.

When we use variables other than x and y to define an unknown quantity, we must change the names of the axes of the graph as well.

Example 4.60

Solution

ⓐ

• Let $h=\mathrm{the number of hamburgers}.$

• $\,c=\mathrm{the number of cookies}$

• To find the system of equations translate the information.

• The calories from hamburgers at 240 calories each, plus the calories from cookies at 160 calories each

• must be more that 800.

• $240h+160c\ge{}800$

• $\,\mathrm{The amount spent on hamburgers at \$1.40 each, plus the amount spent on cookies at \$0.50 each}$

• $\,\mathrm{must be no more than \$5.00.}$

• $1.40h+0.50c\le{}5$

• $\,\mathrm{The number of hamburgers must be greater than or equal to 0.}$

• $h\ge{}0$

• $\,\mathrm{The number of cookies must be greater than or equal to 0.}$

• $c\ge{}0$

• We have our system of equations. $\,\mathrm{\{}\begin{array}{l} 240h+160c\ge{}800 \\ 1.40h+0.50c\le{}5 \\ h\ge{}0 \\ c\ge{}0 \\ \end{array}$

ⓑ Since $h>=0$ and $c>=0$ (both are greater than or equal to) all solutions will be in the first quadrant. As a result, our graph shows only quadrant one.

Graph $1.40h+0.50c\le{}5.$ The boundary line is $1.40h+0.50c=5.$ We test (0, 0) and it makes the inequality true. We shade the side of the line that includes (0, 0).

The solution of the system is the region of the graph that is shaded the darkest. The boundary line sections that border the darkly shaded section are included in the solution as are the points on the x-axis from (5, 0) to (10, 0).

ⓒ To determine if 3 hamburgers and 1 cookie would meet Omar’s criteria, we see if the point (3, 2) is in the solution region. It is, so Omar might choose to eat 3 hamburgers and 1 cookie.

ⓓ To determine if 2 hamburgers and 4 cookies would meet Omar’s criteria, we see if the point (2, 4) is in the solution region. It is, Omar might choose to eat 2 hamburgers and 4 cookies.

We could also test the possible solutions by substituting the values into each inequality.

Section 4.7 Exercises

Section 4.7 Exercises

Determine Whether an Ordered Pair is a Solution of a System of Linear Inequalities

In the following exercises, determine whether each ordered pair is a solution to the system.

Solve a System of Linear Inequalities by Graphing

In the following exercises, solve each system by graphing.

Solve Applications of Systems of Inequalities

In the following exercises, translate to a system of inequalities and solve.

Writing Exercises

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?

1. Determine whether the ordered pair is a solution to the system: $\mathrm{\{}\begin{array}{l} x-5y>10 \\ 2x+3y>-2 \\ \end{array}.$

ⓐ $(3,-1)$ ⓑ $(6,-3)$

2. Determine whether the ordered pair is a solution to the system: $\mathrm{\{}\begin{array}{l} y>4x-2 \\ 4x-y<20 \\ \end{array}.$

ⓐ $(-2,1)$ ⓑ $(4,-1)$

3. Solve the system by graphing: $\mathrm{\{}\begin{array}{l} y<3x+2 \\ y>-x-1 \\ \end{array}.$

1. Solve the system by graphing: $\mathrm{\{}\begin{array}{l} y<-\frac{1}{2}x+3 \\ y<3x-4 \\ \end{array}.$

2. Solve the system by graphing: $\mathrm{\{}\begin{array}{l} x+y\le{}2 \\ y\ge\frac{2}{3}x-1 \\ \end{array}.$

3. Solve the system by graphing: $\mathrm{\{}\begin{array}{l} 3x-2y\le{}6 \\ y>-\frac{1}{4}x+5 \\ \end{array}.$

1. Solve the system by graphing: $\mathrm{\{}\begin{array}{l} y\ge{}3x-2 \\ y<-1 \\ \end{array}.$

2. Solve the system by graphing: $\mathrm{\{}\begin{array}{l} x>-4 \\ x-2y\ge-4 \\ \end{array}.$

3. Solve the system by graphing: $\mathrm{\{}\begin{array}{l} 3x-2y\ge{}12 \\ y\ge\frac{3}{2}x+1 \\ \end{array}.$

1. Solve the system by graphing: $\mathrm{\{}\begin{array}{l} x+3y>8 \\ y<-\frac{1}{3}x-2 \\ \end{array}.$

2. Solve the system by graphing: $\mathrm{\{}\begin{array}{l} y\ge{}3x+1 \\ -3x+y\ge-4 \\ \end{array}.$

3. Solve the system by graphing: $\mathrm{\{}\begin{array}{l} y\le-\frac{1}{4}x+2 \\ x+4y\le{}4 \\ \end{array}.$

1. A trailer can carry a maximum weight of 160 pounds and a maximum volume of 15 cubic feet. A microwave oven weighs 30 pounds and has 2 cubic feet of volume, while a printer weighs 20 pounds and has 3 cubic feet of space.

ⓐ Write a system of inequalities to model this situation.ⓑ Graph the system.ⓒ Could 4 microwaves and 2 printers be carried on this trailer?ⓓ Could 7 microwaves and 3 printers be carried on this trailer?

2. Mary needs to purchase supplies of answer sheets and pencils for a standardized test to be given to the juniors at her high school. The number of the answer sheets needed is at least 5 more than the number of pencils. The pencils cost $2 and the answer sheets cost$1. Mary’s budget for these supplies allows for a maximum cost of $400.

ⓐ Write a system of inequalities to model this situation.ⓑ Graph the system.ⓒ Could Mary purchase 100 pencils and 100 answer sheets?ⓓ Could Mary purchase 150 pencils and 150 answer sheets?

3. Tenison needs to eat at least an extra 1,000 calories a day to prepare for running a marathon. He has only $25 to spend on the extra food he needs and will spend it on$0.75 donuts which have 360 calories each and $2 energy drinks which have 110 calories.

ⓐ Write a system of inequalities that models this situation.ⓑ Graph the system.ⓒ Can he buy 8 donuts and 4 energy drinks and satisfy his caloric needs?ⓓ Can he buy 1 donut and 3 energy drinks and satisfy his caloric needs?

1. Philip’s doctor tells him he should add at least 1,000 more calories per day to his usual diet. Philip wants to buy protein bars that cost $1.80 each and have 140 calories and juice that costs$1.25 per bottle and have 125 calories. He doesn’t want to spend more than $12.

ⓐ Write a system of inequalities that models this situation.ⓑ Graph the system.ⓒ Can he buy 3 protein bars and 5 bottles of juice?ⓓ Can he buy 5 protein bars and 3 bottles of juice?

2. $\mathrm{\{}\begin{array}{l} 4x-y<10 \\ -2x+2y>-8 \\ \end{array}$

ⓐ $(5,-2)$ ⓑ $(-1,3)$

3. $\mathrm{\{}\begin{array}{l} y<\frac{3}{2}x+3 \\ \frac{3}{4}x-2y<5 \\ \end{array}$

ⓐ $(-4,-1)$ ⓑ $\mathrm{(8, 3)}$

1. $\mathrm{\{}\begin{array}{l} 6x-5y<20 \\ -2x+7y>-8 \\ \end{array}$

ⓐ $\mathrm{(1, -3)}$ ⓑ $\mathrm{(-4, 4)}$

2. $\mathrm{\{}\begin{array}{l} y<-2x+2 \\ y\ge-x-1 \\ \end{array}$

3. $\mathrm{\{}\begin{array}{l} y\ge-\frac{2}{3}x+2 \\ y>2x-3 \\ \end{array}$

1. $\mathrm{\{}\begin{array}{l} x+2y<4 \\ y<x-2 \\ \end{array}$

2. $\mathrm{\{}\begin{array}{l} 2x+4y\ge{}8 \\ y\le\frac{3}{4}x \\ \end{array}$

3. $\mathrm{\{}\begin{array}{l} 3x-2y\le{}6 \\ -4x-2y>8 \\ \end{array}$

1. $\mathrm{\{}\begin{array}{l} 2x+y>-6 \\ -x+2y\ge-4 \\ \end{array}$

2. $\mathrm{\{}\begin{array}{l} x-3y>4 \\ y\le-1 \\ \end{array}$

3. $\mathrm{\{}\begin{array}{l} y\le-\frac{2}{3}x+5 \\ x\ge{}3 \\ \end{array}$

1. $\mathrm{\{}\begin{array}{l} y\le-\frac{1}{2}x+3 \\ y<1 \\ \end{array}$

2. $\mathrm{\{}\begin{array}{l} -3x+5y>10 \\ x>-1 \\ \end{array}$

3. $\mathrm{\{}\begin{array}{l} x\le-1 \\ y\ge{}3 \\ \end{array}$

1. $\mathrm{\{}\begin{array}{l} x-3y\ge{}6 \\ y>\frac{1}{3}x+1 \\ \end{array}$

2. $\mathrm{\{}\begin{array}{l} -3x+6y>12 \\ 4y\le{}2x-4 \\ \end{array}$

3. $\mathrm{\{}\begin{array}{l} y\ge\frac{1}{2}x-1 \\ -2x+4y\ge{}4 \\ \end{array}$

1. $\mathrm{\{}\begin{array}{l} y\ge{}3x-1 \\ -3x+y>-4 \\ \end{array}$

2. $\mathrm{\{}\begin{array}{l} y<\frac{3}{4}x-2 \\ -3x+4y<7 \\ \end{array}$

3. Jake does not want to spend more than $50 on bags of fertilizer and peat moss for his garden. Fertilizer costs$2 a bag and peat moss costs $5 a bag. Jake’s van can hold at most 20 bags.

ⓐ Write a system of inequalities to model this situation. ⓑ Graph the system. ⓒ Can he buy 15 bags of fertilizer and 4 bags of peat moss? ⓓ Can he buy 10 bags of fertilizer and 10 bags of peat moss?

1. Juan is studying for his final exams in chemistry and algebra. he knows he only has 24 hours to study, and it will take him at least three times as long to study for algebra than chemistry.

ⓐ Write a system of inequalities to model this situation. ⓑ Graph the system. ⓒ Can he spend 4 hours on chemistry and 20 hours on algebra? ⓓ Can he spend 6 hours on chemistry and 18 hours on algebra?

2. Mark is attempting to build muscle mass and so he needs to eat at least an additional 80 grams of protein a day. A bottle of protein water costs $3.20 and a protein bar costs$1.75. The protein water supplies 27 grams of protein and the bar supplies 16 gram. If he has $10 dollars to spend

ⓐ Write a system of inequalities to model this situation. ⓑ Graph the system. ⓒ Could he buy 3 bottles of protein water and 1 protein bar? ⓓ Could he buy no bottles of protein water and 5 protein bars?

3. Mark is increasing his exercise routine by running and walking at least 4 miles each day. His goal is to burn a minimum of 1500 calories from this exercise. Walking burns 270 calories/mile and running burns 650 calories.

ⓐ Write a system of inequalities to model this situation. ⓑ Graph the system. ⓒ Could he meet his goal by walking 3 miles and running 1 mile? ⓓ Could he meet his goal by walking 2 miles and running 2 miles?

1. Graph the system $\mathrm{\{}\begin{array}{l} x+2y\le{}6 \\ y\ge-\frac{1}{2}x-4 \\ \end{array}.$ What does the solution mean?