1. Learning objectives

Learning objectives

Solve applications with linear inequalities

2. 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.

3. Example 3.53

Example 3.53

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?

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 $r=$ 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, $a>x$ has the same meaning as $x<a$ .	$\begin{array}{rll} 5,625 & \ge & 3r \\ 1,875 & \ge & r \\ r & \le & 1,875 \\ \end{array}$
Step 6. Check the answer in the problem and make sure it makes sense. A maximum rent of $1,875 seems reasonable for an income of$ 5,625.
Step 7. Answer the question with a complete sentence.	The maximum rent is $1,875.

Table

3.18

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.

4. Example 3.54

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 $n=$ 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. $254.12n\le{}4,000$
Step 5. Solve the inequality. But $n$ must be a whole number of tablets, so round to 15.	$n\le{}15.74$ $n\le{}15$
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.

Table

3.19

5. Example 3.55

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 $s=$ 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.	$500+0.12s>925$
Step 5. Solve the inequality.	$\begin{array}{rll} 0.12s & > & 425 \\ s & > & 3,541.\overset{---}{66} \\ \end{array}$
Step 6. Check the answer in the problem and make sure it makes sense. If we round the total sales up to $4,000, we see that$ 500+0.12(4,000)=980 $, which is more than$ 925.
Step 7. Answer the question with a complete sentence.	The total sales must be more than $3,541.67.

Table

3.20

6. Example 3.56

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 $m=$ 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. $75+0.25m\le{}200$
Step 5. Solve the inequality.	$\begin{array}{rll} 0.25m & \le & 125 \\ m & \le & 500\,\mathrm{miles} \\ \end{array}$
Step 6. Check the answer in the problem and make sure it makes sense. Yes, $75+0.25(500)=200$ .
Step 7. Write a sentence that answers the question.	Sergio and Lizeth can travel 500 miles and still stay on budget.

Table

3.21

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.

7. Example 3.57

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

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.

8. Example 3.58

Example 3.58

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?

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 $h=$ 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. $ $350+$ 375+ $60(3)\le$ 500+ $15h$
Step 5. Solve the inequality.	$\begin{array}{rll} 905 & \le & 500+15h \\ 405 & \le & 15h \\ 27 & \le & h \\ h & \ge & 27 \\ \end{array}$
Step 6. Check the answer in the problem and make sure it makes sense. We substitute 27 into the inequality. $\begin{array}{l} \,905\le{}500+15h \\ \,905\le{}500+15(27) \\ \,905\le{}905 \\ \end{array}$
Step 7. Write a sentence that answers the question.	Brenda must babysit at least 27 hours.

9. Section 3.6 Exercises

Section 3.6 Exercises

Solve Applications with Linear Inequalities

In the following exercises, solve.

Everyday Math

Writing Exercises

ⓐ 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?

10. Practice 1

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?

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?

11. Practice 2

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?

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?

12. Practice 3

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?

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?

13. Practice 4

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?

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?

14. Practice 5

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?

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?

15. Practice 6

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?

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?

16. Practice 7

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?

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?

17. Practice 8

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?

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?

18. Practice 9

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?

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?

19. Practice 10

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?

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?

20. Practice 11

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?

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?

21. Practice 12

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?

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?

22. Practice 13

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?

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?

23. Practice 14

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.