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Solve Applications with Linear Inequalities

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

Try it.

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		\(\begin{array}{lll}5,625 & \ge & 3r \\ \\ 1,875 & \ge & r \\ r & \le & 1,875\end{array}\)
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.

Condensed — the full section is in OpenStax Elementary Algebra 2e.

Key Concepts

  • Solving inequalities
    1. Read the problem.
    2. Identify what we are looking for.
    3. Name what we are looking for. Choose a variable to represent that quantity.
    4. Translate. Write a sentence that gives the information to find it. Translate into an inequality.
    5. Solve the inequality.
    6. Check the answer in the problem and make sure it makes sense.
    7. Answer the question with a complete sentence.

Chapter 3 Review Exercises

Approach Word Problems with a Positive Attitude

In the following exercises, reflect on your approach to word problems.

Try it.

How has your attitude towards solving word problems changed as a result of working through this chapter? Explain.

Solution

answers will vary

Try it.

Did the problem-solving strategy help you solve word problems in this chapter? Explain.

Use a Problem-Solving Strategy for Word Problems

In the following exercises, solve using the problem-solving strategy for word problems. Remember to write a complete sentence to answer each question.

Try it.

Three-fourths of the people at a concert are children. If there are 87 children, what is the total number of people at the concert?

Solution

116

Try it.

There are nine saxophone players in the band. The number of saxophone players is one less than twice the number of tuba players. Find the number of tuba players.

Solve Number Problems

In the following exercises, solve each number word problem.

Try it.

The sum of a number and three is forty-one. Find the number.

Solution

38

Try it.

Twice the difference of a number and ten is fifty-four. Find the number.

Try it.

One number is nine less than another. Their sum is negative twenty-seven. Find the numbers.

Solution

\(-18,-9\)

Try it.

One number is eleven more than another. If their sum is increased by seventeen, the result is 90. Find the numbers.

Try it.

One number is two more than four times another. Their sum is \(-13.\) Find the numbers.

Solution

\(-3,-10\)

Try it.

The sum of two consecutive integers is \(-135.\) Find the numbers.

Try it.

Find three consecutive integers whose sum is \(-141.\)

Solution

\(-48,-47,-46\)

Try it.

Find three consecutive even integers whose sum is 234.

Try it.

Find three consecutive odd integers whose sum is 51.

Solution

15, 17, 19

Try it.

Koji has $5,502 in his savings account. This is $30 less than six times the amount in his checking account. How much money does Koji have in his checking account?

Condensed — the full section is in OpenStax Elementary Algebra 2e.

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

  1. Write as an inequality: x is at least 30.
    If you missed this problem, review .

    Onthul het antwoord

    \(x\ge 30\)

  2. Solve \(8-3y<41.\)
    If you missed this problem, review .

    Onthul het antwoord

    \(y>-11\)

  3. 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?

    Onthul het antwoord
    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    		\(\begin{array}{lll}5,625 & \ge & 3r \\ \\ 1,875 & \ge & r \\ r & \le & 1,875\end{array}\)
    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.

  4. 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?

    Onthul het antwoord

    There can be no more than 20 boxes.

  5. 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?

    Onthul het antwoord

    A maximum of 14 people can safely ride in the elevator.

  6. 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?

    Onthul het antwoord
    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.

  7. 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?

    Onthul het antwoord

    seven packs

  8. 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?

    Onthul het antwoord

    11 people

  9. 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?

    Onthul het antwoord
    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.\(\begin{array}{lll}0.12s & > & 425 \\ s & > & 3,541.\overset{\text{—}}{66}\end{array}\)
    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.

  10. 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?

    Onthul het antwoord

    at least $4,000,000

  11. 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?

    Onthul het antwoord

    at least $1,200,000

  12. 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?

    Onthul het antwoord
    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.\(\begin{array}{lll}0.25m & \le & 125 \\ m & \le & 500\ \text{miles}\end{array}\)
    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.

  13. 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?

    Onthul het antwoord

    no more than 106 text messages

  14. 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?

    Onthul het antwoord

    no more than 76 therms

  15. 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?

    Onthul het antwoord
    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 \(j=\) 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 \(60(90)-1,100\), or \(\$4,300\). This is more than \(\$4,000\).
    Step 7. Write a sentence that answer the question.Elliot must work at least 85 jobs.

  16. 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?

    Onthul het antwoord

    at least 96 hours

  17. 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?

    Onthul het antwoord

    at least 1,380 invitations

  18. 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?

    Onthul het antwoord
    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.
    \(\text{\$}350+\text{\$}375+\text{\$}60(3)\le \text{\$}500+\text{\$}15h\)
    Step 5. Solve the inequality.\(\begin{array}{lll} \\ 905 & \le & 500+15h \\ 405 & \le & 15h \\ 27 & \le & h \\ h & \ge & 27\end{array}\)
    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.

  19. 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?

    Onthul het antwoord

    at least 23 hours

  20. 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?

    Onthul het antwoord

    at least 20 driveways

  21. 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?

    Onthul het antwoord

    15 children

  22. 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?

  23. 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?

    Onthul het antwoord

    12 people

  24. 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?

  25. 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?

    Onthul het antwoord

    five drinks

  26. 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?

  27. 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?

    Onthul het antwoord

    31 aprons

  28. 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?

  29. 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?

    Onthul het antwoord

    $15,000

  30. 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?

  31. 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?

    Onthul het antwoord

    $400,000

  32. 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?

  33. 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?

    Onthul het antwoord

    16 ccf

  34. 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?

  35. 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?

    Onthul het antwoord

    nine movies

  36. 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?

  37. 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?

    Onthul het antwoord

    48 cut & styles

  38. 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?

  39. 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?

    Onthul het antwoord

    44 meals

  40. 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?

Symbols used here

\leq,\ \geq
less/greater than or equal
Inequalities that allow equality; < and > exclude it.
\pm
plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.
\neq
not equal
The two sides are different.
\sqrt{x},\ \sqrt[n]{x}
square root, n-th root
The non-negative number whose square (n-th power) is x.
|x|
absolute value / modulus
Distance from zero: |−3| = 3. For a complex number, distance from the origin.
i
imaginary unit
i² = −1.
\log_b x,\ \ln x
logarithm, natural log
The exponent b must be raised to for x; ln uses base e.
\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}
number sets
Naturals, integers, rationals, reals, complex numbers.

How to: Solve Applications with Linear Inequalities

  1. Solve applications with linear inequalities

Questions people ask

What does it mean to solve an equation?

To find every value of the unknown that makes both sides equal. Each step is an operation applied to both sides that keeps the solution set the same, until the unknown stands alone.

Why do I sometimes get two answers?

A quadratic can cross the axis twice, so it can have two solutions. A degree-n polynomial has up to n. The graph shows where each one comes from.

How do I know whether to factor or use the quadratic formula?

Try factoring for a few seconds: look for two numbers that multiply to a·c and add to b. If nothing obvious appears, the discriminant b² − 4ac tells you how many real roots there are, and the formula finds them without guessing.

Probeer je eigen

Parts of this page are adapted from OpenStax Elementary Algebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

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