Worksheet: Multistep Inequalities in Real-World Problems

In this worksheet, we will practice formulating a multistep inequality in a real-world context and solving it to get the range of values satisfying that inequality.

Q1:

Sameh ran 600 metres in 90 seconds, while Amira was at least 9 seconds ahead of him. Write an inequality to represent the time Amira finished the 600-meter run.

  • A π‘₯ ≀ 9 9
  • B π‘₯ β‰₯ 8 1
  • C π‘₯ β‰₯ 9 9
  • D π‘₯ ≀ 8 1
  • E π‘₯ < 9 0

Q2:

A moving company charges a flat rate of $150 and an additional $5 for each box. If a taxi service charged $20, for how many boxes would it be cheaper to move with the taxi service than with the moving company?

  • Aonly at 10 boxes
  • Bmore than 10 boxes
  • Cmore than 6 boxes
  • Dless than 10 boxes
  • Eless than 6 boxes

Q3:

In a board game, Maged scored 22, 11, 23, 19, and 17 points in five turns. Find the minimum number of points he must score in the sixth turn to have an average of at least 17 points.

Q4:

Michael wants to purchase a precision balance for $340. He has already saved $113 and can save $28 every week. Write an inequality that can be used to determine the number of weeks left for Michael to save at least $340.

  • A 1 1 3 π‘₯ + 2 8 β‰₯ 3 4 0
  • B 2 8 π‘₯ + 1 1 3 ≀ 3 4 0
  • C 1 1 3 π‘₯ + 2 8 ≀ 3 4 0
  • D 2 8 π‘₯ + 1 1 3 β‰₯ 3 4 0
  • E 2 8 π‘₯ + 1 1 3 > 3 4 0

Q5:

Sophia finds two landscape gardeners online: the first charges a fixed fee of $20 per job plus $15 per hour for labor, while the second charges a fixed fee of $90 but only $5 per hour for labor. After how many hours will the second gardener be cheaper than the first?

Q6:

You are choosing between two different window-washing companies. The first charges $ 5 per window, while the second charges a base fee of $ 4 0 plus $ 3 per window. How many windows would you need to have for the second company to be preferable?

  • Amore than 15 windows
  • Bmore than 5 windows
  • Cmore than 8 windows
  • Dmore than 20 windows
  • EMore than 10 windows

Q7:

Michael finds two plumbers online: the first charges $20 per hour of labor, while the second charges a fixed charge of $40 per job plus an hourly labor charge of $15. After how many hours will the second plumber be cheaper than the first?

Q8:

Shady earns $6.75 an hour waiting on tables. He is saving money to buy a new video game which costs $27. Write an inequality to find how many hours Shady must work to buy the game, and then solve it.

  • A 2 7 π‘₯ ≀ 6 . 7 5 , π‘₯ ≀ 1 4
  • B 6 . 7 5 π‘₯ ≀ 2 7 , π‘₯ ≀ 4
  • C 2 7 π‘₯ β‰₯ 6 . 7 5 , π‘₯ β‰₯ 1 4
  • D 6 . 7 5 π‘₯ β‰₯ 2 7 , π‘₯ β‰₯ 4
  • E 6 . 7 5 π‘₯ > 2 7 , π‘₯ > 4

Q9:

Adel has a total of $200 and wants to buy some Blu-ray discs. Given that Blu-ray discs cost $18.75 each, and he must save at least $65, write an inequality that can be used to find how many Blu-ray discs he can buy, and then determine the maximum number of Blu-ray discs he can buy.

  • A 1 8 . 7 5 π‘₯ βˆ’ 2 0 0 β‰₯ 6 5 , 14 Blu-ray discs
  • B 2 0 0 βˆ’ 1 8 . 7 5 π‘₯ ≀ 6 5 , 7 Blu-ray discs
  • C 1 8 . 7 5 π‘₯ βˆ’ 2 0 0 ≀ 6 5 , 14 Blu-ray discs
  • D 2 0 0 βˆ’ 1 8 . 7 5 π‘₯ β‰₯ 6 5 , 7 Blu-ray discs
  • E 1 8 . 7 5 π‘₯ βˆ’ 2 0 0 ≀ 6 5 , 7 Blu-ray discs

Q10:

Mona has saved $31 in her piggy bank and her sister Rania has saved $36. If Rania saves $6 a week and Mona saves $9 a week, after how many weeks will Mona have saved more than Rania?

Q11:

You are choosing between two different prepaid cell phone plans. The first plan charges a rate of 26 cents per minute. The second plan charges a monthly fee of $19.95 plus 11 cents per minute. How many minutes would you have to use in a month in order for the second plan to be preferable?

  • Amore than 54 minutes
  • Bmore than 1.33 minutes
  • Cmore than 0.54 minutes
  • Dmore than 133 minutes
  • Emore than 37 minutes

Q12:

A cell phone company offers the following two plans:

  • Plan A: $20 per month and $1 for every one hundred texts.
  • Plan B: $50 per month with free, unlimited texts.

How many texts would you need to send per month for plan B to save you money?

  • Amore than 30
  • Bless than 3 0 0 0
  • Cless than 30
  • Dmore than 3 0 0 0

Q13:

A car rental company offers two plans for renting a car.

  • Plan A: $30per day and $ 0.18 per mile
  • Plan B: $50per day with free unlimited mileage

How many miles would you need to drive for plan B to save you money?

  • Amore than444.44 mi
  • Bless than111.11 mi
  • Cless than444.44 mi
  • Dmore than 111.11 mi

Q14:

A cell phone company offers the following two plans:

  • Plan A: $15 per month and $2 for every 300 texts.
  • Plan B: $25 per month and $0.50 for every 100 texts.

How many texts would you need to send per month for plan B to save you money?

  • Amore than 60
  • Bless than 6 0 0 0
  • Cless than 60
  • Dmore than 6 0 0 0

Q15:

When hired at a new job selling jewelry, you are given two pay options:

  • Option A: a base salary of $ 1 7 0 0 0 a year with a commission of 1 2 % of your sales
  • Option B: a base salary of $ 2 0 0 0 0 a year with a commission of 5 % of your sales

How many dollars’ worth of jewelry would you need to sell for option A to produce a larger income?

  • Amore than $ 3 0 0 0
  • Bmore than $ 1 7 6 4 7 . 0 6
  • Cmore than $ 2 1 7 6 4 7 . 0 6
  • Dmore than $ 4 2 8 5 7 . 1 4

Q16:

A basic cellular package costs $20 per month for 60 min of calling, with an additional charge of $0.30 per minute beyond that time. The cost formula would be 𝐢 = $ 2 0 + 0 . 3 0 ( π‘₯ βˆ’ 6 0 ) , where π‘₯ is the number of minutes used. Given that your monthly bill must be less than $50, find the maximum number of minutes you can use in a month.

Q17:

William and Daniel are competing on a quiz app. William has 400 points and is losing 2 points per minute; Daniel has 250 points and is winning 10 points per minute.

Write an inequality which can be used to find π‘š , the amount of time for which William has no fewer points than Daniel.

  • A 2 0 0 βˆ’ 3 π‘š ≀ 2 5 0 + 1 5 π‘š
  • B 4 0 0 + 2 π‘š < 2 5 0 + 1 0 π‘š
  • C 4 0 0 + 3 π‘š > 2 5 0 + 1 0 π‘š
  • D 4 0 0 βˆ’ 2 π‘š β‰₯ 2 5 0 + 1 0 π‘š
  • E 2 0 0 βˆ’ 2 π‘š β‰₯ 2 5 0 + 1 5 π‘š

Use your inequality to find the time when Daniel catches up with William. Assume that points are won or lost at a constant rate.

  • A 1 2 1 2 minutes
  • B 6 minutes
  • C 14 minutes
  • D 1 2 1 3 minutes
  • E 1 5 1 2 minutes

Q18:

The cost of producing β€œdoomflots” is $325 for equipment plus another $32 for materials and labour for each one produced. How many β€œdoomflots” need to be produced such that the average cost per β€œdoomflot” is less than $50?

  • A10
  • B9
  • C18
  • D19

Q19:

Anthony and Charlotte were saving their allowances. Anthony has started with $150 in his account and deposited $20 at the end of every month; Charlotte started with $50 in her account and deposited $32 at the end of every month.

Write an inequality that can be used to find π‘š , the number of months for which there was more money in Anthony’s account than in Charlotte’s.

  • A 1 4 0 + 3 0 π‘š < 5 0 + 6 4 π‘š
  • B 2 0 βˆ’ 1 5 0 π‘š > 5 0 + 3 2 π‘š
  • C 1 5 0 + 2 0 π‘š < 5 0 + 3 2 π‘š
  • D 1 5 0 + 2 0 π‘š > 5 0 + 3 2 π‘š
  • E 7 5 βˆ’ 2 0 π‘š β‰₯ 5 0 + 6 4 π‘š

Use your inequality to find π‘š .

Q20:

When hired at a new job selling electronics, you are given two pay options:

  • Option A: a base salary of $ 1 0 0 0 0 a year with a commission of 9 % of your sales.
  • Option B: a base salary of $ 2 0 0 0 0 a year with a commission of 4 % of your sales.

How many dollars’ worth of electronics would you need to sell for Option A to produce a larger income?

  • Amore than $ 2 3 0 7 6 9 . 2 3 worth of electronics
  • Bmore than $ 6 0 0 0 0 0 worth of electronics
  • Cmore than $ 7 6 9 2 3 . 0 8 worth of electronics
  • Dmore than $ 2 0 0 0 0 0 worth of electronics

Q21:

When hired at a new job selling electronics, you are given two pay options:

  • Option A: a base salary of $ 2 0 0 0 0 a year with a commission of 1 2 % of your sales
  • Option B: a base salary of $ 2 6 0 0 0 a year with a commission of 3 % of your sales

How many dollars’ worth of electronics would you need to sell for option A to produce a larger income?

  • Amore than $ 4 0 0 0 0 worth of electronics
  • Bmore than $ 5 1 1 1 1 1 . 1 1 worth of electronics
  • Cmore than $ 3 0 6 6 6 6 . 6 7 worth of electronics
  • Dmore than $ 6 6 6 6 6 . 6 7 worth of electronics

Q22:

You just got a new job in retail and you are given two options for your earnings:

Option 1: Base salary of $15,000 per year plus a commission of 1 4 % of your sales.

Option 2: Base salary of $18,000 per year plus a commission of 1 0 % of your sales.

How much money in sales would you need to make in order for Option 1 to yield a higher income than Option 2?

  • AMore than $137,500
  • BMore than $12,500
  • CMore than $18,000
  • DMore than $75,000

Q23:

Sameh wants to spend some of his birthday money to buy stationery. He buys a pencil case and some crayons to go in it. A pencil case costs $2 and crayons cost 75 cents each. Sameh has $10.

Write an inequality that can be used to find 𝑛 , the number of crayons he can buy.

  • A 2 + 0 . 7 5 𝑛 < 1 0
  • B 1 0 + 0 . 5 𝑛 ≀ 2
  • C 1 0 + 0 . 5 𝑛 > 2
  • D 2 + 0 . 7 5 𝑛 ≀ 1 0
  • E 1 0 + 0 . 7 5 𝑛 β‰₯ 2

Solve your inequality to find the maximum number of crayons that Sameh can buy.

Q24:

Matthew needs to buy some clothes. The store’s parking lot has the shown sign outside.

Write an inequality for , the time in hours, that Matthew can park if he has only $8.25 in cash.

  • A
  • B
  • C
  • D
  • E

Given that you must pay for whole hours of parking, use your inequality to find the maximum time that Matthew can park.

  • A 6 hours
  • B 5 hours
  • C 7 hours
  • D 12 hours
  • E 9 hours

Q25:

When the sum of a number and twelve is multiplied by four, the result is more than when three times the number is subtracted from eleven.

Write an inequality to represent the statement above. Let π‘₯ represent the number.

  • A 4 ( π‘₯ + 1 2 ) ≀ 7 + 5 π‘₯
  • B 4 ( π‘₯ + 1 2 ) β‰₯ 1 1 βˆ’ 3 π‘₯
  • C 4 ( π‘₯ + 1 2 ) > 3 π‘₯ βˆ’ 1 1
  • D 4 ( π‘₯ + 1 2 ) > 1 1 βˆ’ 3 π‘₯
  • E 4 ( π‘₯ + 1 2 ) < 3 π‘₯ βˆ’ 1 1

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