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Algebra 1

2.13.2 Solving Problems with Inequalities in Two Variables

Algebra 12.13.2 Solving Problems with Inequalities in Two Variables
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Activity

Here are three types of situations. Answer the questions about each situation.

Bank Accounts

For questions 1 – 4, use the following scenario:

A customer opens a checking account and a savings account at a bank. A maximum of $600 will be deposited into the two accounts, some in the checking account and some in the savings account. (They might not deposit all of it and keep some of the money as cash.)
1.

Write an inequality to describe the constraints. Specify what each variable represents.

2.

Rewrite your inequality using slope-intercept form. Then, use the graphing tool or technology outside the course. Graph the inequality that represents this scenario using the Desmos tool below.

Hint: To restrict the xx and yy values to the constraints of the problem, enter {0≤x≤600}{0≤y≤600}{0≤x≤600}{0≤y≤600} after your equation.

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3.

Name one solution to the inequality and explain what it represents in that situation.

4.

If the customer deposits $200 in their checking account, what can you say about the amount they deposit in their savings account?

For questions 5 – 8, use the following scenario:

A customer opens a checking account and a savings account at a bank. The bank requires a minimum balance of $50 in the savings account. It does not matter how much money is kept in the checking account.
5.

Write an inequality to describe the constraints. Specify what each variable represents

6.

Rewrite your inequality using slope-intercept form. Then, use the graphing tool or technology outside the course. Graph the inequality that represents this scenario using the Desmos tool below.

Hint: To restrict the xx values to the constraints of the problem (the checking account should never have less than $0), enter {x≥0}{x≥0} after your equation.

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7.

Name one solution to the inequality and explain what it represents in that situation.

8.

If the customer deposits no money in the checking account but is able to maintain both accounts without penalty, what can you say about the amount deposited in the savings account?

Concert Tickets

For questions 1 – 4, use the following scenario:

Two kinds of tickets to an outdoor concert were sold: lawn tickets and seat tickets. Fewer than 400 tickets in total were sold.
1.

Write an inequality to describe the constraints. Specify what each variable represents.

2.

Rewrite your inequality using slope-intercept form. Then, use the graphing tool or technology outside the course. Graph the inequality that represents this scenario using the Desmos tool below.

Hint: To restrict the xx and yy values to the constraints of the problem, enter {0≤x}{0≤y}{0≤x}{0≤y} after your equation (because you cannot have a negative number of tickets).

Access multimedia content
3.

Name one solution to the inequality and explain what it represents in that situation.

4.

If you know that exactly 100 lawn tickets were sold, what can you say about the number of seat tickets?

For questions 5 – 8, use the following scenario:

Lawn tickets cost $30 each and seat tickets cost $50 each. The organizers made at least $14,000 from ticket sales.
5.

Write an inequality to describe the constraints. Specify what each variable represents.

6.

Rewrite your inequality using slope-intercept form. Then, use the graphing tool or technology outside the course. Graph the inequality that represents this scenario using the Desmos tool below.

Hint: To restrict the xx and yy values to the constraints of the problem, enter {0≥x}{0≥y}{0≥x}{0≥y} after your equation (because you cannot have a negative number of tickets).

Access multimedia content
7.

Name one solution to the inequality and explain what it represents in that situation.

8.

If you know that exactly 200 seat tickets were sold, what can you say about the number of lawn tickets?

Advertising Packages

For questions 1 – 4, use the following scenario:

An advertising agency offers two packages for small businesses who need advertising services. A basic package includes only design services. A premium package includes design and promotion. The agency's goal is to sell at least 60 packages in total.
1.

Write an inequality to describe the constraints. Specify what each variable represents.

2.

Rewrite your inequality using slope-intercept form. Then, use the graphing tool or technology outside the course. Graph the inequality that represents this scenario using the Desmos tool below.

Hint: To restrict the x and y values to the constraints of the problem, enter {x≥0}{y≥0}{x≥0}{y≥0} after your equation (because the business will not have a negative number of packages).

Access multimedia content
3.

Name one solution to the inequality and explain what it represents in that situation.

4.

If the agency sells exactly 45 basic packages, what can you say about the number of premium packages it needs to sell to meet its goal?

For questions 5 – 8, use the following scenario:

The basic advertising package has a value of $1000 and the premium package has a value of $2500. The goal of the agency is to sell more than $60,000 worth of small-business advertising packages.
5.

Write an inequality to describe the constraints. Specify what each variable represents.

6.

Rewrite your inequality using slope-intercept form. Then, use the graphing tool or technology outside the course. Graph the inequality that represents this scenario using the Desmos tool below.

Hint: To restrict the xx and yy values to the constraints of the problem, enter {x≥0}{y≥0}{x≥0}{y≥0} after your equation.

Access multimedia content
7.

Name one solution to the inequality and explain what it represents in that situation.

8.

If you know that exactly 10 premium packages were sold, what can you say about the number of basic packages the agency needs to sell to meet its goal?

Video: Solving Problems with Inequalities in Two Variables

Watch the following video to learn more about solving inequalities in two variables.

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Are you ready for more?

Extending Your Thinking

This activity will require a partner and either graph paper or Desmos.

1.

Without letting your partner see it, write an equation of a line so that both the xx-intercepts and the yy-intercepts are each between -3 and 3. Graph your equation.

2.

Still without letting your partner see it, rewrite your equation as an inequality. That is, write an inequality for which your equation is the related equation. In other words, your line should be the boundary between the solution region and non-solution region. Shade the solutions on your graph.

3.

Take turns stating coordinates of points. Your partner will tell you whether your guess is a solution to their inequality. After each partner has stated a point, each may guess what the other's inequality is. If neither guesses correctly, play continues. Use another coordinate system to keep track of your guesses.

What was your process for determining your partner’s inequality?

Self Check

Use graphing technology to solve the following problem.

Jacob charges $20 to mow a lawn in his neighborhood and $100 to paint a room.  

Which combination of jobs earns Jacob more than $1,000?

  1. Jacob mows 5 lawns and paints 9 rooms.
  2. Jacob mows 35 lawns and paints 1 room.
  3. Jacob mows 8 lawns and paints 10 rooms.
  4. Jacob mows 10 lawns and paints 8 rooms.

Additional Resources

Graphing Linear Inequalities in Two Variables to Solve Applications

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.

For questions 1 – 3, use the following scenario to write, graph, and find solutions to an inequality:

Example

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? 

1. Let xx be the number of hours she works at the job in food service and let yy be the number of hours she works tutoring. Write an inequality that would model this situation.

Solution

We let xx be the number of hours she works at the job in food service and let yy 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.

Mathematical expression: 10 times x plus 15 times y is greater than or equal to 240. Above the inequality, labels indicate the term representing 10 times x is the amount earned at the food service job, the addition sign represents plus, and the term representing 15 times y is the amount earned tutoring. The phrase "is at least 240 dollars" represents the label for greater than or equal to 240.

2. Graph the inequality.

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

10x+15y≥24015y≥−10x+240y≥−23x+1610x+15y≥24015y≥−10x+240y≥−23x+16

A graph with a red line sloping down from (0, 16) to (24, 0) on a coordinate plane. The region above the line and within the first quadrant is shaded light red. The line is solid so it is included in the answer region.

3. Find three ordered pairs (x,y)(x,y) that would be solutions to the inequality. Then, explain what that means for Hilaria.

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

insert alt text

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.

Try it

Try It: Graphing Linear Inequalities in Two Variables to Solve Applications

For questions 1 – 3, use the following scenario to write, graph, and find solutions to an inequality:

Elijah works two part-time jobs. One is at a grocery store that pays $10 an hour and the other is babysitting for $13 per hour. Between the two jobs, Elijah wants to earn at least $260 a week. How many hours does Elijah need to work at each job to earn at least $260? 
1.

Let xx be the number of hours he works at the grocery store and let yy be the number of hours he works babysitting. Write an inequality that would model this situation.

2.

Graph the inequality.

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3.

Find three ordered pairs (x,y)(x,y) that would be solutions to the inequality. Then, explain what that means for Elijah.

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