Solve Applications with Linear Inequalities

By the end of this section, you will be able to: solve applications with linear inequalities.

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. Emma got a new job and will have to move. Her monthly income will be $5,625. 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?

Identify what we are looking for: the highest rent Emma will qualify for. Name what we are looking for: let $r =$ the rent.

Translate into an inequality. Emma’s monthly income must be at least three times the rent:

5{,}625 \geq 3r

Solve the inequality — divide both sides by 3:

1{,}875 \geq r

\leq 1{,}875

Check: a maximum rent of $1,875 seems reasonable for an income of $5,625.

Answer the question: the maximum rent is $1,875.

Alan is loading a pallet with boxes that each weigh 45 pounds. The pallet can safely support no more than 900 pounds. How many boxes can he safely load onto the pallet? Write the answer as the maximum whole number of boxes.

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? Write the answer as the maximum whole number of people.

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. 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?

Name what we are looking for: let $n =$ the number of tablets.

Translate into an inequality. $254.12 times the number of tablets is no more than $4,000:

254.12n \leq 4{,}000

Solve the inequality — divide both sides by 254.12:

n \leq 15.74\ldots

But $n$ must be a whole number of tablets, so round down to 15:

n \leq 15

Check: 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.

Answer the question: Dawn can buy a maximum of 15 tablets.

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? Write the answer as a whole number.

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? Write the answer as a whole number.

Example. 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?

Identify what we are looking for: the total sales needed for his variable pay option to exceed the fixed amount of $925. Name what we are looking for: let $s =$ the total sales.

Translate into an inequality. Remember to convert the percent to a decimal. $500 plus 12% of sales is more than $925:

500 + 0.12s > 925

Solve the inequality — subtract 500 from both sides, then divide by 0.12:

0.12s > 425

s > 3{,}541.\overline{66}

Check: if we round the total sales up to $4,000, we see that $500 + 0.12(4{,}000) = 980$ , which is more than $925.

Answer the question: the total sales must be more than $3,541.67.

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? Write the answer as an inequality for the total sales s.

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? Write the answer as an inequality for the total sales s.

Example. 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?

Name what we are looking for: let $m =$ the number of miles.

Translate into an inequality. $75 plus $0.25 times the number of miles is less than or equal to $200:

75 + 0.25m \leq 200

Solve the inequality — subtract 75 from both sides, then divide by 0.25:

0.25m \leq 125

m \leq 500 \text{ miles}

Check: $75 + 0.25(500) = 200$ , which checks.

Answer the question: Sergio and Lizeth can travel 500 miles and still stay on budget.

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? Write the answer as a whole number.

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? Write the answer as a whole number.

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. 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?

Name what we are looking for: let $j =$ the number of jobs.

Translate into an inequality. $60 times the number of jobs minus $1,100 is at least $4,000:

60j - 1{,}100 \geq 4{,}000

Solve the inequality — add 1,100 to both sides, then divide by 60:

60j \geq 5{,}100

j \geq 85 \text{ jobs}

Check: if Elliot did 90 jobs, his profit would be $60(90) - 1{,}100$ , or $4,300. This is more than $4,000.

Answer the question: Elliot must work at least 85 jobs.

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? Write the answer as a whole number.

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? Write the answer as a whole number.

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. Brenda’s best friend is having a destination wedding and the event will require 3 nights in a hotel. 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?

Name what we are looking for: let $h =$ the number of hours.

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:

350 + 375 + 60(3) \leq 500 + 15h

Solve the inequality — combine the constants on the left, subtract 500 from both sides, then divide by 15:

905 \leq 500 + 15h

405 \leq 15h

27 \leq h

h \geq 27

Check: substitute 27 into the inequality:

905 \leq 500 + 15(27)

905 \leq 905

Answer the question: Brenda must babysit at least 27 hours.

Malik is planning a 6-day summer vacation trip. He has $840 in savings, and he earns $45 per hour tutoring. The trip will cost him $525 for airfare, $780 for food and sightseeing, and $95 per night for the hotel (6 nights). How many hours must he tutor to have enough money to pay for the trip? Write the answer as a whole number.

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 (10 nights). 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? Write the answer as a whole number.

Key terms

profit — the money that remains when the expenses have been subtracted from the money earned. commission — pay calculated as a percent of sales, often combined with a fixed base amount.

This section is adapted from Elementary Algebra 2e, Section 3.6: Solve Applications with Linear Inequalities by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Changes: omitted the Be Prepared quiz, Self Check checklist, and end-of-section practice exercises; and converted the practice problems (“Try Its”) into interactive exercises with instant feedback.

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