3.6 Solve Applications with Linear Inequalities

Before you get started, take this readiness quiz.

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

Solve $8 - 3 y < 41 .$
If you missed this problem, review the reference.

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?

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?

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?

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?

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?

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?

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?

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?

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?

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?

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?

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?

Solve.

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?

Solve.

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?

Solve.

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?

Solve.

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?

Solve.

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?

Solve.

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?

Solve.

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?

Solve.

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?

Solve.

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?

Solve.

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 ord…

Solve.

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?

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?

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.

Reflect on your approach to word problems.

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

Solve using the problem-solving strategy for word problems. Remember to write a complete sentence to answer each question.

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?

Solve each number word problem.

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

Solve each number word problem.

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

Solve each number word problem.

One number is two more than four times another. Their sum is $−13 .$ Find the numbers.

Solve each number word problem.

Find three consecutive integers whose sum is $−141 .$

Solve each number word problem.

Find three consecutive odd integers whose sum is 51.

Translate and solve.

What number is 67% of 250?

Translate and solve.

12.5% of what number is 20?

Translate and solve.

What percent of 125 is 150?

Solve.

The bill for Dino’s lunch was $19.45. He wanted to leave 20% of the total bill as a tip. How much should the tip be?

Solve.

Dolores bought a crib on sale for $350. The sale price was 40% of the original price. What was the original price of the crib?

Solve.

Angel’s got a raise in his annual salary from $55,400 to $56,785. Find the percent increase.

Solve.

Winston deposited $3,294 in a bank account with interest rate 2.6%. How much interest was earned in 5 years?

Solve.

Jaime’s refrigerator loan statement said he would pay $1,026 in interest for a 4-year loan at 13.5%. How much did Jaime borrow to buy the refrigerator?

Find the sale price.

The original price of a handbag was $84. Carole bought it on sale for $21 off.

Find ⓐ the amount of discount and ⓑ the sale price.

Emmett bought a pair of shoes on sale at 40% off from an original price of $138.

Find ⓐ the amount of discount and ⓑ the discount rate. (Round to the nearest tenth of a percent, if needed.)

Zack bought a printer for his office that was on sale for $380. The original price of the printer was $450.

Find ⓐ the amount of the mark-up and ⓑ the list price.

Nga and Lauren bought a chest at a flea market for $50. They re-finished it and then added a 350% mark-up.

Solve each coin word problem.

Francie has $4.35 in dimes and quarters. The number of dimes is five more than the number of quarters. How many of each coin does she have?

Solve each coin word problem.

Paulette has $140 in $5 and $10 bills. The number of $10 bills is one less than twice the number of $5 bills. How many of each does she have?

Solve each ticket or stamp word problem.

A church luncheon made $842. Adult tickets cost $10 each and children’s tickets cost $6 each. The number of children was 12 more than twice the number of adults. How many of each ticket were sold?

Solve each ticket or stamp word problem.

125 tickets were sold for the jazz band concert for a total of $1,022. Student tickets cost $6 each and general admission tickets cost $10 each. How many of each kind of ticket were sold?

Solve each ticket or stamp word problem.

Ana spent $4.06 buying stamps. The number of $0.41 stamps she bought was five more than the number of $0.26 stamps. How many of each did she buy?

Solve each mixture word problem.

Marquese is making 10 pounds of trail mix from raisins and nuts. Raisins cost $3.45 per pound and nuts cost $7.95 per pound. How many pounds of raisins and how many pounds of nuts should Marquese use for the trail mix to cost him $6.96 per pound?

Solve each mixture word problem.

Shawn has $15,000 to invest. She will put some of it into a fund that pays 4.5% annual interest and the rest in a certificate of deposit that pays 1.8% annual interest. How much should she invest in each account if she wants to earn 4.05% annual interest on the total amount?

Solve using triangle properties.

The measures of two angles of a triangle are 22 and 85 degrees. Find the measure of the third angle.

Solve using triangle properties.

A triangular road sign has base 30 inches and height 40 inches. What is its area?

Solve using triangle properties.

One angle of a triangle is $30 °$ more than the smallest angle. The largest angle is the sum of the other angles. Find the measures of all three angles.

Solve using triangle properties.

The measure of the smallest angle in a right triangle is $45 °$ less than the measure of the next larger angle. Find the measures of all three angles.

Solve using rectangle properties.

The length of a rectangle is 36 feet and the width is 19 feet. Find the ⓐ perimeter ⓑ area.

Solve using rectangle properties.

The area of a rectangle is 2356 square meters. The length is 38 meters. What is the width?

Solve using rectangle properties.

The length of a rectangle is 12 cm more than the width. The perimeter is 74 cm. Find the length and the width.

Solve.

When Gabe drives from Sacramento to Redding it takes him 2.2 hours. It takes Elsa 2 hours to drive the same distance. Elsa’s speed is seven miles per hour faster than Gabe’s speed. Find Gabe’s speed and Elsa’s speed.

Solve.

Two busses leave Amarillo at the same time. The Albuquerque bus heads west on the I-40 at a speed of 72 miles per hour, and the Oklahoma City bus heads east on the I-40 at a speed of 78 miles per hour. How many hours will it take them to be 375 miles apart?

Solve.

At 6:30, Devon left her house and rode her bike on the flat road until 7:30. Then she started riding uphill and rode until 8:00. She rode a total of 15 miles. Her speed on the flat road was three miles per hour faster than her speed going uphill. Find Devon’s speed on the flat road and riding uphill.

Solve.

Julianne has a weekly food budget of $231 for her family. If she plans to budget the same amount for each of the seven days of the week, what is the maximum amount she can spend on food each day?

Solve.

Briana has been offered a sales job in another city. The offer was for $42,500 plus 8% of her total sales. In order to make it worth the move, Briana needs to have an annual salary of at least $66,500. What would her total sales need to be for her to move?

Solve.

Costa is an accountant. During tax season, he charges $125 to do a simple tax return. His expenses for buying software, renting an office, and advertising are $6,000. How many tax returns must he do if he wants to make a profit of at least $8,000?

Four-fifths of the people on a hike are children. If there are 12 children, what is the total number of people on the hike?

The sum of two consecutive odd integers is $−96 .$ Find the numbers.

Humberto’s hourly pay increased from $16.25 to $17.55. Find the percent increase.

Dotty bought a freezer on sale for $486.50. The original price of the freezer was $695. Find ⓐ the amount of discount and ⓑ the discount rate.

At a concert, $1,600 in tickets were sold. Adult tickets were $9 each and children’s tickets were $4 each. If the number of adult tickets was 30 less than twice the number of children’s tickets, how many of each kind were sold?

The measure of one angle of a triangle is twice the measure of the smallest angle. The measure of the third angle is 14 more than the measure of the smallest angle. Find the measures of all three angles.

Use the Pythagorean Theorem to find the length of the missing side. Round to the nearest tenth, if necessary.

Two planes leave Dallas at the same time. One heads east at a speed of 428 miles per hour. The other plane heads west at a speed of 382 miles per hour. How many hours will it take them to be 2,025 miles apart?

Use the Pythagorean Theorem to find the length of the missing side. Round to the nearest tenth, if necessary.

Chloe has a budget of $800 for costumes for the 18 members of her musical theater group. If all the costumes are the same price, what is the maximum she can spend for each costume?