Learning Objectives
By the end of this section, you will be able to:
- Solve applications with linear inequalities
Be Prepared 3.6
Before you get started, take this readiness quiz.
- Write as an inequality: x is at least 30.
If you missed this problem, review Example 2.77. - Solve
If you missed this problem, review Example 2.73.
Solve Applications with Linear Inequalities
Many real-life situations require us to solve inequalities. In fact, inequality applications are so common that we often do not even realize we are doing algebra. For example, how many gallons of gas can be put in the car for $20? Is the rent on an apartment affordable? Is there enough time before class to go get lunch, eat it, and return? How much money should each family member’s holiday gift cost without going over budget?
The method we will use to solve applications with linear inequalities is very much like the one we used when we solved applications with equations. We will read the problem and make sure all the words are understood. Next, we will identify what we are looking for and assign a variable to represent it. We will restate the problem in one sentence to make it easy to translate into an inequality. Then, we will solve the inequality.
Example 3.53
Emma got a new job and will have to move. Her monthly income will be $5,265. To qualify to rent an apartment, Emma’s monthly income must be at least three times as much as the rent. What is the highest rent Emma will qualify for?
Solution
| Step 1. Read the problem. | |
| Step 2. Identify what we are looking for. | the highest rent Emma will qualify for |
| Step 3. Name what we are looking for.
Choose a variable to represent that quantity. |
Let the rent. |
| Step 4. Translate into an inequality. First write a sentence that gives the information to find it. |
Emma's monthly income must be at least three times the rent. |
| Step 5. Solve the inequality.
Remember, has the same meaning as . |
|
| Step 6. Check the answer in the problem and make sure it makes sense.
A maximum rent of $1,755 seems reasonable for an income of $5,625. |
|
| Step 7. Answer the question with a complete sentence. | The maximum rent is $1,755. |
Try It 3.105
Alan is loading a pallet with boxes that each weighs 45 pounds. The pallet can safely support no more than 900 pounds. How many boxes can he safely load onto the pallet?
Try It 3.106
The elevator in Yehire’s apartment building has a sign that says the maximum weight is 2,100 pounds. If the average weight of one person is 150 pounds, how many people can safely ride the elevator?
Sometimes an application requires the solution to be a whole number, but the algebraic solution to the inequality is not a whole number. In that case, we must round the algebraic solution to a whole number. The context of the application will determine whether we round up or down. To check applications like this, we will round our answer to a number that is easy to compute with and make sure that number makes the inequality true.
Example 3.54
Dawn won a mini-grant of $4,000 to buy tablet computers for her classroom. The tablets she would like to buy cost $254.12 each, including tax and delivery. What is the maximum number of tablets Dawn can buy?
Solution
| Step 1. Read the problem. | |
| Step 2. Identify what we are looking for. | the maximum number of tablets Dawn can buy |
| Step 3. Name what we are looking for.
Choose a variable to represent that quantity. |
Let the number of tablets. |
| Step 4. Translate. Write a sentence that gives the information to find it. Translate into an inequality. |
$254.12 times the number of tablets is no more than $4,000.
|
| Step 5. Solve the inequality.
But must be a whole number of tablets, so round to 15. |
|
| Step 6. Check the answer in the problem and make sure it makes sense.
Rounding down the price to $250, 15 tablets would cost $3,750, while 16 tablets would be $4,000. So a maximum of 15 tablets at $254.12 seems reasonable. |
|
| Step 7. Answer the question with a complete sentence. | Dawn can buy a maximum of 15 tablets. |
Try It 3.107
Angie has $20 to spend on juice boxes for her son’s preschool picnic. Each pack of juice boxes costs $2.63. What is the maximum number of packs she can buy?
Try It 3.108
Daniel wants to surprise his girlfriend with a birthday party at her favorite restaurant. It will cost $42.75 per person for dinner, including tip and tax. His budget for the party is $500. What is the maximum number of people Daniel can have at the party?
Example 3.55
Pete works at a computer store. His weekly pay will be either a fixed amount, $925, or $500 plus 12% of his total sales. How much should his total sales be for his variable pay option to exceed the fixed amount of $925?
Solution
| Step 1. Read the problem. | |
| Step 2. Identify what we are looking for. | the total sales needed for his variable pay option to exceed the fixed amount of $925 |
| Step 3. Name what we are looking for.
Choose a variable to represent that quantity. |
Let the total sales. |
| Step 4. Translate. Write a sentence that gives the information to find it. Translate into an inequality. Remember to convert the percent to a decimal. |
|
| Step 5. Solve the inequality. | |
| Step 6. Check the answer in the problem and make sure it makes sense.
If we round the total sales up to $4,000, we see that , which is more than $925. |
|
| Step 7. Answer the question with a complete sentence. | The total sales must be more than $3,541.67. |
Try It 3.109
Tiffany just graduated from college and her new job will pay her $20,000 per year plus 2% of all sales. She wants to earn at least $100,000 per year. For what total sales will she be able to achieve her goal?
Try It 3.110
Christian has been offered a new job that pays $24,000 a year plus 3% of sales. For what total sales would this new job pay more than his current job which pays $60,000?
Example 3.56
Sergio and Lizeth have a very tight vacation budget. They plan to rent a car from a company that charges $75 a week plus $0.25 a mile. How many miles can they travel and still keep within their $200 budget?
Solution
| Step 1. Read the problem. | |
| Step 2. Identify what we are looking for. | the number of miles Sergio and Lizeth can travel |
| Step 3. Name what we are looking for.
Choose a variable to represent that quantity. |
Let the number of miles. |
| Step 4. Translate. Write a sentence that gives the information to find it. Translate into an inequality. |
$75 plus 0.25 times the number of miles is less than or equal to $200. |
| Step 5. Solve the inequality. | |
| Step 6. Check the answer in the problem and make sure it makes sense.
Yes, . |
|
| Step 7. Write a sentence that answers the question. | Sergio and Lizeth can travel 500 miles and still stay on budget. |
Try It 3.111
Taleisha’s phone plan costs her $28.80 a month plus $0.20 per text message. How many text messages can she use and keep her monthly phone bill no more than $50?
Try It 3.112
Rameen’s heating bill is $5.42 per month plus $1.08 per therm. How many therms can Rameen use if he wants his heating bill to be a maximum of $87.50?
A common goal of most businesses is to make a profit. Profit is the money that remains when the expenses have been subtracted from the money earned. In the next example, we will find the number of jobs a small businessman needs to do every month in order to make a certain amount of profit.
Example 3.57
Elliot has a landscape maintenance business. His monthly expenses are $1,100. If he charges $60 per job, how many jobs must he do to earn a profit of at least $4,000 a month?
Solution
| Step 1. Read the problem. | |
| Step 2. Identify what we are looking for. | the number of jobs Elliot needs |
| Step 3. Name what we are looking for. Choose a variable to represent it. | Let the number of jobs. |
| Step 4. Translate Write a sentence that gives the information to find it. | $60 times the number of jobs minus $1,100 is at least $4,000. |
| Translate into an inequality. | |
| Step 5. Solve the inequality. | |
| Step 6. Check the answer in the problem and make sure it makes sense. If Elliot did 90 jobs, his profit would be , or . This is more than . |
|
| Step 7. Write a sentence that answer the question. | Elliot must work at least 85 jobs. |
Try It 3.113
Caleb has a pet sitting business. He charges $32 per hour. His monthly expenses are $2,272. How many hours must he work in order to earn a profit of at least $800 per month?
Try It 3.114
Felicity has a calligraphy business. She charges $2.50 per wedding invitation. Her monthly expenses are $650. How many invitations must she write to earn a profit of at least $2,800 per month?
Sometimes life gets complicated! There are many situations in which several quantities contribute to the total expense. We must make sure to account for all the individual expenses when we solve problems like this.
Example 3.58
Brenda’s best friend is having a destination wedding and the event will last 3 days. Brenda has $500 in savings and can earn $15 an hour babysitting. She expects to pay $350 airfare, $375 for food and entertainment and $60 a night for her share of a hotel room. How many hours must she babysit to have enough money to pay for the trip?
Solution
| Step 1. Read the problem. | |
| Step 2. Identify what we are looking for. | the number of hours Brenda must babysit |
| Step 3. Name what we are looking for.
Choose a variable to represent that quantity. |
Let the number of hours. |
| Step 4. Translate. Write a sentence that gives the information to find it. Translate into an inequality. |
The expenses must be less than or equal to the income.
The cost of airfare plus the cost of food and entertainment and the hotel bill must be less than or equal to the savings plus the amount earned babysitting. |
| Step 5. Solve the inequality. | |
| Step 6. Check the answer in the problem and make sure it makes sense.
We substitute 27 into the inequality. |
|
| Step 7. Write a sentence that answers the question. | Brenda must babysit at least 27 hours. |
Try It 3.115
Malik is planning a 6-day summer vacation trip. He has $840 in savings, and he earns $45 per hour for tutoring. The trip will cost him $525 for airfare, $780 for food and sightseeing, and $95 per night for the hotel. How many hours must he tutor to have enough money to pay for the trip?
Try It 3.116
Josue wants to go on a 10-day road trip next spring. It will cost him $180 for gas, $450 for food, and $49 per night for a motel. He has $520 in savings and can earn $30 per driveway shoveling snow. How many driveways must he shovel to have enough money to pay for the trip?
Section 3.6 Exercises
Practice Makes Perfect
Solve Applications with Linear Inequalities
In the following exercises, solve.
Mona is planning her son’s birthday party and has a budget of $285. The Fun Zone charges $19 per child. How many children can she have at the party and stay within her budget?
Carlos is looking at apartments with three of his friends. They want the monthly rent to be no more than $2360. If the roommates split the rent evenly among the four of them, what is the maximum rent each will pay?
A water taxi has a maximum load of 1,800 pounds. If the average weight of one person is 150 pounds, how many people can safely ride in the water taxi?
Marcela is registering for her college classes, which cost $105 per unit. How many units can she take to have a maximum cost of $1,365?
Arleen got a $20 gift card for the coffee shop. Her favorite iced drink costs $3.79. What is the maximum number of drinks she can buy with the gift card?
Teegan likes to play golf. He has budgeted $60 next month for the driving range. It costs him $10.55 for a bucket of balls each time he goes. What is the maximum number of times he can go to the driving range next month?
Joni sells kitchen aprons online for $32.50 each. How many aprons must she sell next month if she wants to earn at least $1,000?
Ryan charges his neighbors $17.50 to wash their car. How many cars must he wash next summer if his goal is to earn at least $1,500?
Keshad gets paid $2,400 per month plus 6% of his sales. His brother earns $3,300 per month. For what amount of total sales will Keshad’s monthly pay be higher than his brother’s monthly pay?
Kimuyen needs to earn $4,150 per month in order to pay all her expenses. Her job pays her $3,475 per month plus 4% of her total sales. What is the minimum Kimuyen’s total sales must be in order for her to pay all her expenses?
Andre has been offered an entry-level job. The company offered him $48,000 per year plus 3.5% of his total sales. Andre knows that the average pay for this job is $62,000. What would Andre’s total sales need to be for his pay to be at least as high as the average pay for this job?
Nataly is considering two job offers. The first job would pay her $83,000 per year. The second would pay her $66,500 plus 15% of her total sales. What would her total sales need to be for her salary on the second offer be higher than the first?
Jake’s water bill is $24.80 per month plus $2.20 per ccf (hundred cubic feet) of water. What is the maximum number of ccf Jake can use if he wants his bill to be no more than $60?
Kiyoshi’s phone plan costs $17.50 per month plus $0.15 per text message. What is the maximum number of text messages Kiyoshi can use so the phone bill is no more than $56.50?
Marlon’s TV plan costs $49.99 per month plus $5.49 per first-run movie. How many first-run movies can he watch if he wants to keep his monthly bill to be a maximum of $100?
Kellen wants to rent a banquet room in a restaurant for her cousin’s baby shower. The restaurant charges $350 for the banquet room plus $32.50 per person for lunch. How many people can Kellen have at the shower if she wants the maximum cost to be $1,500?
Moshde runs a hairstyling business from her house. She charges $45 for a haircut and style. Her monthly expenses are $960. She wants to be able to put at least $1,200 per month into her savings account order to open her own salon. How many “cut & styles” must she do to save at least $1,200 per month?
Noe installs and configures software on home computers. He charges $125 per job. His monthly expenses are $1,600. How many jobs must he work in order to make a profit of at least $2,400?
Katherine is a personal chef. She charges $115 per four-person meal. Her monthly expenses are $3,150. How many four-person meals must she sell in order to make a profit of at least $1,900?
Melissa makes necklaces and sells them online. She charges $88 per necklace. Her monthly expenses are $3745. How many necklaces must she sell if she wants to make a profit of at least $1,650?
Five student government officers want to go to the state convention. It will cost them $110 for registration, $375 for transportation and food, and $42 per person for the hotel. There is $450 budgeted for the convention in the student government savings account. They can earn the rest of the money they need by having a car wash. If they charge $5 per car, how many cars must they wash in order to have enough money to pay for the trip?
Cesar is planning a 4-day trip to visit his friend at a college in another state. It will cost him $198 for airfare, $56 for local transportation, and $45 per day for food. He has $189 in savings and can earn $35 for each lawn he mows. How many lawns must he mow to have enough money to pay for the trip?
Alonzo works as a car detailer. He charges $175 per car. He is planning to move out of his parents’ house and rent his first apartment. He will need to pay $120 for application fees, $950 for security deposit, and first and last months’ rent at $1,140 per month. He has $1,810 in savings. How many cars must he detail to have enough money to rent the apartment?
Eun-Kyung works as a tutor and earns $60 per hour. She has $792 in savings. She is planning an anniversary party for her parents. She would like to invite 40 guests. The party will cost her $1,520 for food and drinks and $150 for the photographer. She will also have a favor for each of the guests, and each favor will cost $7.50. How many hours must she tutor to have enough money for the party?
Everyday Math
Maximum Load on a Stage In 2014, a high school stage collapsed in Fullerton, California, when 250 students got on stage for the finale of a musical production. Two dozen students were injured. The stage could support a maximum of 12,750 pounds. If the average weight of a student is assumed to be 140 pounds, what is the maximum number of students who could safely be on the stage?
Maximum Weight on a Boat In 2004, a water taxi sank in Baltimore harbor and five people drowned. The water taxi had a maximum capacity of 3,500 pounds (25 people with average weight 140 pounds). The average weight of the 25 people on the water taxi when it sank was 168 pounds per person. What should the maximum number of people of this weight have been?
Wedding Budget Adele and Walter found the perfect venue for their wedding reception. The cost is $9,850 for up to 100 guests, plus $38 for each additional guest. How many guests can attend if Adele and Walter want the total cost to be no more than $12,500?
Shower Budget Penny is planning a baby shower for her daughter-in-law. The restaurant charges $950 for up to 25 guests, plus $31.95 for each additional guest. How many guests can attend if Penny wants the total cost to be no more than $1,500?
Writing Exercises
Find your last month’s phone bill and the hourly salary you are paid at your job. (If you do not have a job, use the hourly salary you would realistically be paid if you had a job.) Calculate the number of hours of work it would take you to earn at least enough money to pay your phone bill by writing an appropriate inequality and then solving it.
Find out how many units you have left, after this term, to achieve your college goal and estimate the number of units you can take each term in college. Calculate the number of terms it will take you to achieve your college goal by writing an appropriate inequality and then solving it.
Self Check
ⓐ 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?