Unlocking How to Solve a System of Inequalities: A Comprehensive Guide

Every individual benefits from the efficient use of their resources such as time and money. To use such resources effectively, several inequalities might need to be solved simultaneously. Together, these inequalities form a system. This lesson will teach how to write systems of inequalities and how to solve them using graphs.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Challenge

Matching Graphs With Inequalities

Each of the following graphs represents the solution set of a certain inequality.

Five different coordinate planes with a line and a region shaded

Pair each graph with its corresponding inequality.

Explore

Analyzing Plane Regions Formed by Two Inequalities

In the following applet, each of the solution sets of the inequalities presented previously can be viewed on the same coordinate plane. Plane regions are formed once two inequalities are drawn. Only two can be drawn at a time.

A coordinate plane and 5 buttons to display five different inequalities. Only two inequalities can be seen at the same time

Think about what each of these regions — overlapping, shaded, and unshaded — represent.

Discussion

System of Linear Inequalities

As seen earlier, when two or more inequalities are graphed on the same coordinate plane, their solution sets may overlap. In these cases, the set of all inequalities being solved simultaneously forms a system of inequalities.

Concept

System of Inequalities

A system of inequalities is a set of two or more inequalities involving the same variables. For example, the set formed by the following two inequalities is a system.

{y \leq - 0.5 x + 3 y > x

The solution set of a system of inequalities is the set of all ordered pairs that satisfy all the inequalities in the system simultaneously. The ordered pair

(0, 1),

for example, is a solution to the above system.

{1 \leq - 0.5 (0) + 3 1 > 0 ✓ ✓

Usually, systems of inequalities are solved by graphing each inequality on the same coordinate plane. When the inequalities in a system are graphed, the coordinate plane is divided into different regions. These regions provide an insight into the determination of the solution set. To see what these regions represent for the aforementioned system, move point

P .

Of the regions formed, the overlapping region represents the solution set of the system.

Method

Solving a System of Linear Inequalities Graphically

A system of linear inequalities can be solved by graphing all inequalities on the same coordinate plane and then finding the region of intersection, if any. For example, consider the following system.

{x + y < 7 x + 2 y \leq 10 (I) (II)

To solve the system graphically, these three steps can be followed.

Write the Inequalities in Slope-Intercept Form

To simplify the process of graphing the inequalities, start by writing them in slope-intercept form.

{x + y < 7 x + 2 y \leq 10 (I) (II)

SubIneq

$(I):$ $LHS - x < RHS - x$

{y < - x + 7 x + 2 y \leq 10

SubIneq

$(II):$ $LHS - x \leq RHS - x$

{y < - x + 7 2 y \leq - x + 10

DivIneq

$(II):$ $LHS / 2 \leq RHS / 2$

{y < - x + 7 y \leq - 0.5 x + 5

Graph the Inequalities

To graph the inequalities, begin with writing the corresponding boundary lines. The boundary line of the first inequality is

y = - x + 7 .

Inequality I : Boundary Line I : y < - x + 7 y = - x + 7

Since the inequality is strict, the boundary line should be dashed and, in this case, the shaded region is the one below the line.

Drawing y=-x+7 dashed and the region below it is shaded

Similarly, the boundary line of the second inequality is

y = - 0.5 x + 5 .

Inequality II : Boundary Line II : y < - 0.5 x + 5 y = - 0.5 x + 5

Since the inequality is non-strict, its boundary line is solid. In addition, the region to be shaded is the one below the line. This inequality will be graphed on the same coordinate plane.

Drawing y=-0.5x+5 solid and the region below it is shaded

Find the Overlapping Region

Notice that there is a region where the solution sets of the inequalities overlap. All the points in this region satisfy both inequalities simultaneously. Therefore, the overlapping region is the solution set of the system. In the next graph, only the common region is shaded.

Since the boundary lines in their entirety are not part of the solution set, they can be cropped to show only the edges of the overlapping region, or the exceeding parts can be drawn with lower opacity.

Keep in mind that if there is no overlapping region, the system has no solution.

Example

Buying Gifts Without Going Over Budget

Jordan, feeling jolly, is thinking about giving gifts to her teammates — there are $30$ players. Shopping at a stationery store, she decides it is best to buy some fancy ballpoint and fountain pens. She wants to spend less than $$ 240$ and is now unsure whether to give all or only some of her teammates a gift.

A ballpoint pen labeled for 6 dollars and a fountain pen labeled for 12 dollars

Let $x$ and $y$ be the number of ballpoint and fountain pens Jordan will buy, respectively.

a Help Jordan and write a system of inequalities that represents this situation.

b Graph the solution set of the system.

c If Jordan wants to buy the maximum number of fountain pens she can under her budget, what is the maximum number of teammates that could get a gift? How many of them will get a ballpoint pen?

Answer

{x + y \leq 30 6 x + 12 y < 240

b Graph:

c A maximum of

20

teammates could get a gift. Of that number, only one would get a ballpoint pen.

Hint

a Recall that Jordan can buy at most

30

gifts and that the total cost has to be less than

$ 240 .

b To graph the solution set of the system of inequalities, start by writing the inequalities in slope-intercept form. Note that only points with non-negative and integer coordinates are meaningful.

c Of the points in the solution set, which one has the greatest

y -

coordinate?

Solution

a Since Jordan has

30

teammates, she can buy at most

30

gifts. In other words, the sum of the number of ballpoint pens

x

and the number of fountain pens

y

can be at most

30 .

This leads to writing the first inequality.

x + y \leq 30

Jordan does not want to go over budget and wants to spend less than

$ 240 .

Therefore, the total cost of the pens has to be less than

240 .

Total Cost < 240

The total cost equals the number of ballpoint pens bought multiplied by its price plus the number of fountain pens bought multiplied by its price. Using the fact that each ballpoint pen costs

$ 6.00

and each fountain pen costs

$ 12.00,

a second inequality can be set.

6 x + 12 y < 240

In consequence, the following system of inequalities models Jordan's situation.

{x + y \leq 30 6 x + 12 y < 240 (I) (II)

b To solve the system written in Part A graphically, both inequalities will be graphed on the same coordinate plane.

{x + y \leq 30 6 x + 12 y < 240 (I) (II)

Start by rewriting each inequality in slope-intercept form. To do so, isolate the

y -

variable in the left-hand side of each inequality.

{x + y \leq 30 6 x + 12 y < 240 (I) (II)

▼

Write in slope-intercept form

SubIneq

$(I):$ $LHS - x \leq RHS - x$

{y \leq - x + 30 6 x + 12 y < 240

SubIneq

$(II):$ $LHS - 6 x < RHS - 6 x$

{y \leq - x + 30 12 y < - 6 x + 240

DivIneq

$(II):$ $LHS / 12 < RHS / 12$

{y \leq - x + 30 y < - \frac{1}{2} x + 20

To graph inequality (I), first, draw the boundary line whose equation is

y = - x + 30 .

Since the inequality is non-strict, the line is solid. In this case, the region to be shaded is the one below the line.

Next, graph inequality (II) on the same coordinate plane. To do so, draw the boundary line whose equation is

y = - \frac{1}{2} x + 20 .

Since the inequality is strict, the line will be dashed. As before, the region to be shaded is the one below the line.

The solution to the system of inequalities is the overlapping region. However, since buying a negative number of pens does not make sense, the negative values should be discarded. Therefore, the shaded region will include but also be limited by the coordinate axes.

Additionally, buying

3.5

ballpoint pens is nonsensical. Therefore, only points with integer coordinates are part of the solution. Consequently, the solution set of the system looks as follows.

c Since

y

represents the number of fountain pens that Jordan will buy, in the solution of the system found in Part B, look for the points with the greatest

y -

coordinate.

As seen, in the shaded region, there are two points with the greatest integer

y -

coordinate —

(0, 19)

and

(1, 19) .

Therefore, Jordan has two options, buying no ballpoint pens and

19

fountain pens or buying one ballpoint pen and

19

fountain pens.

With the first option, $19$ teammates will receive a gift, while $20$ teammates will get a gift with the second option. Therefore, if Jordan buys the maximum number of fountain pens she can, the maximum number of teammates that could get a gift is $20 .$ Of those $20$ teammates, only one would get a ballpoint pen.

Example

Working Two Jobs Without Being Overworked

Jordan enjoyed being able to buy gifts for her friends. Now she wants to save up to buy a family members car. To achieve this goal, she dares to work two jobs this summer — one in a mechanic shop and another in a bakery. She can make

$ 9

per hour in the mechanic shop and

$ 8

per hour in the bakery.

Jordan plans to work every day for at least

4

more hours in the bakery than in the mechanic shop. However, since she does not want to overwork, she does not plan to earn more than

$ 108

per day.

a Let

x

and

y

be the number of hours that Jordan will work in the mechanic shop and in the bakery every day, respectively. Write a system of inequalities that models Jordan's situation.

b Solve the system graphically.

c Last Friday, Jordan worked whole number of hours in both places. What is the minimum amount of money she could have earned last Friday?

Answer

{y \geq x + 4 9 x + 8 y \leq 108

b Graph:

$ 49

Hint

a Note that

y

must be greater than or equal to

x + 4 .

Also, be aware that Jordan plans to earn at most

$ 108

per day.

b Both boundary lines have to be drawn solid, and negative values do not make sense.

c Look for the points with integer coordinates inside the region drawn in Part B. Which of these points is farthest from the line representing the maximum daily earnings? Use the coordinates of that point to determine the amount of money earned.

Solution

a Since Jordan wants to work in the bakery at least

4

more hours than in the mechanic shop,

y

must be greater than or equal to

x

plus

4 .

Bakery y^{↑} \geq ↓ x + 4 \geq Mechanic Shop

On the other hand, it is said that Jordan plans to earn no more than

$ 108 .

That is, her daily income is at most

$ 108 .

Daily Income \leq 108

The daily income equals the hourly rate multiplied by the number of hours worked. Since Jordan has two jobs, her daily income equals

9

multiplied by the number of hours worked in the mechanic shop added to

8

multiplied by the number of hours worked in the bakery.

9 x + 8 y \leq 108

Consequently, the following system of inequalities models Jordan's situation.

{y \geq x + 4 9 x + 8 y \leq 108 (I) (II)

b To solve the system written in Part A graphically, both inequalities will be graphed on the same coordinate plane.

{y \geq x + 4 9 x + 8 y \leq 108 (I) (II)

The first inequality is already in slope-intercept form. However, the other inequality needs to be written in slope-intercept form.

{y \geq x + 4 9 x + 8 y \leq 108 (I) (II)

▼

(II):

Write in slope-intercept form

SubIneq

$(II):$ $LHS - 9 x \leq RHS - 9 x$

{y \geq x + 4 8 y \leq - 9 x + 108

DivIneq

$(II):$ $LHS / 8 \leq RHS / 8$

{y \geq x + 4 y \leq - 1.125 x + 13.5

To graph inequality (I), its boundary line

y = x + 4

will be drawn first. Since the inequality is non-strict, the line is solid. Since

(0, 0)

does not satisfy the inequality, the region to be shaded is the one not containing the origin.

Next, inequality (II) will be graphed on the same coordinate plane. To do so, draw the boundary line, whose equation is

y = - 1.125 x + 13.5 .

Since the inequality is non-strict, this line will also be solid. In this case, the region to be shaded is the one below the line.

The solution to the system of inequalities is the overlapping region. Notice that negative values do not make sense in this context. Therefore, the shaded region will include but also be limited by the

y -

axis. Consequently, the system's solution looks as follows.

c It is given that Jordan worked an integer number of hours in both jobs last Friday. Then, look for the points with integer coordinates inside the shaded region drawn in Part B. Since the points on the

y -

axis represent the case where Jordan did not work in the mechanic shop, these points should not be considered.

Because the second line drawn represents Jordan's maximum possible daily income, the further a point is from this line, the lower the income that the point represents. Of the points, the one that is farthest from the line is

(1, 5)

which represents the case when Jordan worked

1

hour in the mechanic shop and

5

hours in the bakery.

(1, 5) \Rightarrow {👨 🏼 ‍ 🔧 \times 1 hour 👨 🏼 ‍ 🍳 \times 5 hour

With this combination of worked hours, the minimum income that Jordan could have earned last Friday can be found. To do so, substitute

x = 1

and

y = 5

into the expression representing the daily income.

Daily Income = 9 x + 8 y

▼

Substitute coordinates and evaluate

SubstituteII

$x = 1$ , $y = 5$

Daily Income = 9 (1) + 8 (5)

Multiply

Daily Income = 9 + 40

AddTerms

Add terms

Daily Income = 49

Consequently, the minimum amount of money that Jordan could have earned last Friday is

$ 49 .

Example

A Cinema's Expected Attendance and Revenue

Jordan now has a bit of extra cash to spend. She is excited for a new movie release. Ticket prices vary depending on the customer's age —

$ 6

for children and

$ 9

for adults. The cinema expects more than

250

people to attend the premiere but they do not expect a revenue of more than

$ 2700 .

A ticket with the prices: Adults (9 dollars), Children (6 dollars)

Jordan finds herself wanting to solve this problem during her break time at the bakery. Let

x

and

y

be the number of children and adults who attended the movie premiere, respectively.

a Write a system of inequalities that models the given situation.

b Solve the system graphically.

c If the cinema's expectations were met and

150

children attended the premiere, what is the maximum and minimum possible number of adults who could have attended the premiere?

Answer

{x + y > 250 6 x + 9 y \leq 2700

b Graph:

c Maximum:

200

Minimum: $101$

Hint

a The number of attendees is greater than

250 .

The cinema's revenue is less than or equal to

2700 .

b The line representing the attendance is dashed while the other line is solid. In this context, only non-negative and integer values for

x

and

y

make sense.

c Look for the points whose

x -

coordinate is

150

and lie inside the shaded region drawn in Part B. Of those points, what is the maximum integer

y -

coordinate?

Solution

a Since the cinema expects an attendance of more than

250

people, the number of children plus the number of adults must be greater than

250 .

x + y > 250

Also, it is said that the estimated revenue will be no more than

$ 2700 .

Therefore, the revenue is less than or equal to

2700 .

Revenue \leq 2700

The cinema's revenue equals the number of children who attended the premiere multiplied by

6

added to the number of adults who attended the premiere multiplied by

9 .

6 x + 9 y \leq 2700

Consequently, the following system of inequalities models the described situation.

{x + y > 250 6 x + 9 y \leq 2700

b To solve the system written in Part A graphically, both inequalities will be graphed on the same coordinate plane.

{x + y > 250 6 x + 9 y \leq 2700 (I) (II)

First, rewrite each inequality in slope-intercept form.

{x + y > 250 6 x + 9 y \leq 2700 (I) (II)

▼

Write in slope-intercept form

SubIneq

$(I):$ $LHS - x > RHS - x$

{y > - x + 250 6 x + 9 y \leq 2700

SubIneq

$(II):$ $LHS - 6 x \leq RHS - 6 x$

{y > - x + 250 9 y \leq - 6 x + 2700

DivIneq

$(II):$ $LHS / 9 \leq RHS / 9$

{y > - x + 250 y \leq - \frac{6}{9} x + \frac{2 7 0 0}{9}

SimpQuot

$(II):$ Simplify quotient

{y > - x + 250 y \leq - \frac{2}{3} x + 300

To graph inequality (I), draw the boundary line whose equation is

y = - x + 250 .

Since the inequality is strict, the line is dashed. In this case, the region to be shaded is the one above the line.

Next, graph inequality (II) on the same coordinate plane. To do so, draw the boundary line whose equation is

y = - \frac{2}{3} x + 300 .

Since the inequality is not strict, this line will be drawn solid. In this case, the region to be shaded is the one below the line.

The solution to the system of inequalities is the overlapping region. However, because of the context, only non-negative and integer values make sense. Therefore, the region where both inequalities overlap will also be limited by the axes, including them.

Instead of showing each point separately on the graph, because it is difficult to do so, the region where these points lie is shown. However, remember that only the points with integer coordinates are meaningful.

c It is known that the expected conditions have occurred.

More than $250$ people attended the premiere.
Less than $$ 2700$ was earned.

Recall that these two conditions were written as a system of inequalities and solved graphically.

{x + y > 250 6 x + 9 y \leq 2700

The solution set of the system shows all the possible cases satisfying both conditions. Now, knowing that there were

150

children in the premiere eliminates most of the possible cases.

The number of adults who could have attended the premiere can be any point whose

x -

coordinate is

150

and lie inside the shaded region. Of those points,

(150, 200)

has the the maximum integer

y -

coordinate and

(150, 101)

has the minimum

y -

coordinate.

(150, 200) \leftarrow Maximum y - coordinate (150, 101) \leftarrow Minimum y - coordinate

Therefore, the number of adults who could have attended the premiere is any number between

101

and

200,

inclusive.

Example

Analyzing a Road Trip Plan

Finally, Jordan was able to save enough money to buy a used car! She now plans to take a mini-road trip to visit a relative across-state. First, she will pick up her sister Ramsha, who lives in a different city. The travel distance depends on the path she chooses, but the entire route is no less than $990$ kilometers. Jordan would like to drive for a maximum of $8$ hours.

A map with a car and two roads to get to Ramsha's house and two more roads to get to Jordan's parents house.

Jordan plans to drive at $70$ kilometers per hour from her house to Ramsha's, and from there, she plans to increase the speed to $110$ kilometers per hour until reaching her relative's house.

a Let

x

be the time it takes Jordan to drive from her house to Ramsha, and let

y

be the time it takes to drive from Ramsha's house to her relative's house. Write a system of inequalities that models Jordan's travel plan.

b Solve the system graphically.

Answer

{x + y \leq 8 70 x + 110 y \geq 990

b The system has no realistic solution — the regions do not intersect in the first quadrant.

Hint

a The sum of the traveling times cannot be greater than

8 .

When speed is multiplied by time, the distance traveled is obtained. That is,

d = r \cdot t .

Note that the sum of the distances is greater than or equal to

990

kilometers.

b Both boundary lines have to be drawn solid. Note that negative values do not make sense.

Solution

a Since Jodran wants to drive at most for

8

hours, the sum of the times,

x

and

y,

should be at most

8 .

With this, the first inequality can be written.

x + y \leq 8

On the other hand, it is said that the entire route is no less than

990

kilometers, which means that the distance to drive is greater than or equal to

990

kilometers.

Distance to Drive \geq 990

This distance can be written in terms of

x

and

y

by using the speed formula

d = r \cdot t .

According to Jordan's plan, she will drive to Ramsha's house at a rate of

70

kilometers per hour. Multiplying this rate by the time elapsed to arrive at Ramsha's house gives the distance traveled from her house to Ramsha's.

Distance from Jordan ’ s house to Ramsha ’ s ⇓ 70 x

Similarly, multiplying the second rate,

110

kilometers per hour, by the time it takes for the rest of the route, the distance from Ramsha's house to their relative's will be obtained.

Distance from Ramsha ’ s house to their relative ’ s ⇓ 110 y

The sum of these distances equals the distance needed to drive. Thus, a second inequality can be set.

70 x + 110 y \geq 990

These two inequalities need to be satisfied simultaneously. Therefore, the two inequalities together form a system of inequalities that models Jordan's travel plan.

{x + y \leq 8 70 x + 110 y \geq 990

b To solve the system written in Part A graphically, both inequalities will be graphed on the same coordinate plane.

{x + y \leq 8 70 x + 110 y \geq 990 (I) (II)

First, rewrite the inequalities in slope-intercept form.

{x + y \leq 8 70 x + 110 y \geq 990 (I) (II)

▼

(I), (II):

Write in slope-intercept form

SubIneq

$(I):$ $LHS - x \leq RHS - x$

{y \leq - x + 8 70 x + 110 y \geq 990

SubIneq

$(II):$ $LHS - 70 x \geq RHS - 70 x$

{y \leq - x + 8 110 y \geq - 70 x + 990

DivIneq

$(II):$ $LHS / 110 \geq RHS / 110$

{y \leq - x + 8 y \geq - \frac{7 0}{1 1 0} x + \frac{9 9 0}{1 1 0}

SimpQuot

$(II):$ Simplify quotient

{y \leq - x + 8 y \geq - \frac{7}{1 1} x + 9

To graph inequality (I), draw the boundary line

y = - x + 8 .

Since the inequality is not strict, the line is dashed. In this case, the region to be shaded is the one below the line.

Next, graph inequality (II) on the same coordinate plane. To do so, first draw the boundary line

y = - \frac{7}{1 1} x + 9 .

Since the inequality is not strict, this line will also be drawn solid. However, the region to be shaded is the one above the line.

The solution to the system of inequalities is the overlapping region. However, in the context of the given situation,

x

and

y

cannot take negative values as they represent time. Therefore, only solutions in the first quadrant, if any, should be considered.

As seen, the system of inequalities has no realistic solution, which means Jordan's plan is not feasible. She will not be able to meet her travel expectations. For a more realistic plan, she should either increase the traveling speed or increase the maximum time she plans to drive. The second option is the safest, though.

Pop Quiz

Solving Systems of Linear Inequalities Graphically

For each given system of linear inequalities, select the region corresponding to its solution set, if any.

Random systems of linear inequalities and different regions shaded

Closure

Systems With More Than Two Inequalities

When working with a system of inequalities, there is no upper limit to the number of inequalities that can be analyzed. Consider, for example, the following system with three inequalities.

⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ - 3 x + 4 y \geq - 8 2 x + y > - 6 x + 4 y \leq 4 (I) (II) (III)

The steps to solve this system are the same as those used to solve a system with only two inequalities.

Write the inequalities in slope-intercept form.
Graph each inequality in the system.
Find the region where all the graphs overlap.

The inequalities in the given system can be rewritten as follows.

⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ y \geq \frac{3}{4} x - 2 y > - 2 x - 6 y \leq - \frac{1}{4} x + 1 (I) (II) (III)

The second step can be performed with the help of the applet.

Graphing the inequalities y greater than or equal to 3/4x-2, y greater than -2x-6, and y less than or equal to -0.25x+1

As can be seen, there is a portion of the plane where the three regions overlap — the triangle in the middle. Any point in that triangle is in the solution set of the system.

Region enclosed by the lines y=3/4x-2, y=-2x-6, and y=-1/4x+1. Triangle formed in the middle.

Note that in the examples considered throughout this lesson, even though only two inequalities were mentioned, the fact that the

x -

and

y -

values could not be negative is equivalent to considering the inequalities

x \geq 0

and

y \geq 0 .

That is, in those examples, four inequalities were considered together.

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Catch-Up and Review

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