3.7: Solve Applications with Linear Inequalities
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- Jan 21, 2024
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Learning Objectives
By the end of this section, you will be able to:
- Solve applications with linear inequalities
Note
Before you get started, take this readiness quiz.
- Write as an inequality: x is at least 30.
If you missed this problem, review Exercise 2.7.34. - Solve 8−3y<41.
If you missed this problem, review Exercise 2.7.22.
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 3.7.1
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 forStep 3. Name what we are looking for.Let r = rentChoose a variable to represent that quantity.Step 4. Translate into an inequality.Emma’s monthly income must be at leastFirst write a sentence that gives the informationthree times the rent.to find it.Step 5. Solve the inequality.5265≥3rRemember, a>x has the same meaning1755≥ras x<ar≤1755Step 6. Check the answer in the problemand make sure it makes sense.A maximum rent of $1,755 seemsreasonable for an income of $5,265.Step 7. Answer the answer in the problemthe question with acomplete sentence.The maximum rent is $1,755.
Try It 3.7.2
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?
- Answer
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There can be no more than 20 boxes.
Try It 3.7.3
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?
- Answer
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A maximum of 14 people can safely ride in 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 3.7.4
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 buyStep 3. Name what we are looking for.Let n = the number of tablets.Choose a variable to represent that quantity.Step 4. Translate. write a sentence thatgives the information to find it.$254.12 times the number of tablets is nomore than $4000.Translate into an inequality.254.12n≤4000Step 5. Solve the inequality.n≤15.74But n must be a whole number of tablets,so round to 15.n≤15Step 6. Check the answer in the problemand make sure it makes sense.Rounding down the price to $250,15 tablets would cost $3750, while16 tablets would be $4000. So amaximum of 15 tablets at $254.12seems reasonable.Step 7. Answer the answer in the problemthe question with acomplete sentence.Dawn can buy a maximum of 15 tablets.
Try It 3.7.5
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?
- Answer
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seven packs
Try It 3.7.6
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?
- Answer
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11 people
Example 3.7.7
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 payoption to exceed the fixed amount of $925Step 3. Name what we are looking for.Let s = the total sales.Choose a variable to represent that quantity.Step 4. Translate. write a sentence thatgives the information to find it.$500 plus 12% of total sales is more than $925.Translate into an inequality. Remember to500+0.12s>925convert the percent to a decimal.Step 5. Solve the inequality.0.12s>425s>3541.¯66Step 6. Check the answer in the problemand make sure it makes sense.Rounding down the price to $250,15 tablets would cost $3750, whileIf we round the total sales up to$4000, we see that500+0.12(4000) = 980, which is morethan $925.Step 7. Answer the the question with a complete sentence.The total sales must be more than $3541.67
Try It 3.7.8
Tiffany just graduated from college and her new job will pay her $20000 per year plus 2% of all sales. She wants to earn at least $100000 per year. For what total sales will she be able to achieve her goal?
- Answer
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at least $4000000
Try It 3.7.9
Christian has been offered a new job that pays $24000 a year plus 3% of sales. For what total sales would this new job pay more than his current job which pays $60000?
- Answer
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at least $1200000
Example 3.7.10
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 travelStep 3. Name what we are looking for.Let m = the number of miles.Choose a variable to represent that quantity.Step 4. Translate. write a sentence that$75 plus 0.25 times the number of miles isgives the information to find it. less than or equal to $200.Translate into an inequality. 75+25m≤200Step 5. Solve the inequality.0.25m≤125m≤500 milesStep 6. Check the answer in the problemand make sure it makes sense.Yes, 75 + 0.25(500) = 200.Step 7. Answer the the question with a complete sentence.Sergio and Lizeth can travel 500 milesand still stay on budget.
Try It 3.7.11
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?
- Answer
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no more than 106 text messages
Try It 3.7.12
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?
- Answer
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no more than 76 therms
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 3.7.13
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 needsStep 3. Name what we are looking for.Choose a variable to represent itLet j = the number of jobs.Step 4. Translate. write a sentence that$60 times the number of jobs minus $1,100 is at least $4,000.gives the information to find it. less than or equal to $200.Translate into an inequality. 60j−1100≥4000Step 5. Solve the inequality.60j≥5100j≥85 jobsStep 6. Check the answer in the problemand make sure it makes sense.If Elliot did 90 jobs, his profit would be60(90)−1,100,or $4,300. This ismore than $4,000.Step 7. Answer the the question with a complete sentence.Elliot must work at least 85 jobs.
Try It 3.7.14
Caleb has a pet sitting business. He charges $32 per hour. His monthly expenses are $2272. How many hours must he work in order to earn a profit of at least $800 per month?
- Answer
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at least 96 hours
Try It 3.7.15
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 $2800 per month?
- Answer
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at least 1380 invitations
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 3.7.16
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 babysitStep 3. Name what we are looking for.Choose a variable to represent that quantity.Let h = the number of hours.Step 4. Translate. write a sentence thatgives the information to find it.The expenses must be less than or equal tothe income. The cost of airfare plus thecost of food and entertainment and thehotel bill must be less than or equal to the savingsplus the amount earned babysitting.Translate into an inequality. $350+$375+$60(3)≤$500+$15hStep 5. Solve the inequality.905≤500+15h405≤15h27≤hh≥27Step 6. Check the answer in the problemand make sure it makes sense.We substitute 27 into the inequality.905≤500+15h905≤500+15(27)905≤905Step 7. Answer the the question with a complete sentence.Brenda must babysit at least 27 hours.
Try It 3.7.17
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?
- Answer
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at least 23 hours
Try It 3.7.18
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?
- Answer
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at least 20 driveways
Key Concepts
- Solving inequalities
- Read the problem.
- Identify what we are looking for.
- Name what we are looking for. Choose a variable to represent that quantity.
- Translate. Write a sentence that gives the information to find it. Translate into an inequality.
- Solve the inequality.
- Check the answer in the problem and make sure it makes sense.
- Answer the question with a complete sentence.