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Worksheet: Multistep Inequalities in Real-World Problems

Q1:

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

  • Aonly at 10 boxes
  • Bless than 10 boxes
  • Cmore than 6 boxes
  • Dmore than 10 boxes
  • Eless than 6 boxes

Q2:

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

Q3:

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

Q4:

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

Q5:

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

Q6:

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

Q7:

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

Q8:

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

Q9:

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.

Q10:

Matthew 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. Matthew 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 Matthew can buy.

Q11:

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 .

Q12:

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

Q13:

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

Q14:

In a board game, Benjamin 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.

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 hotel caters for large parties and events. They charge $300 for the hall and $15 per person for a lunchtime buffet.

Write an inequality that can be used to find 𝑛 , the number of people who can go to a party that was planned with a budget of $ 1 0 0 0 .

  • A 1 5 + 3 0 0 𝑛 ≀ 1 0 0 0
  • B 3 0 0 + 1 5 𝑛 β‰₯ 1 0 0 0
  • C 1 5 + 1 0 0 0 𝑛 ≀ 3 0 0
  • D 3 0 0 + 1 5 𝑛 ≀ 1 0 0 0
  • E 1 5 + 1 0 0 0 𝑛 β‰₯ 3 0 0

Use your inequality to find the maximum number of people.

Q17:

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

Q18:

Noah 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

Q19:

A candy store has a special offer: if you spend more than $15, you get a free chocolate drink. Gift boxes are $3 each, and chocolates are $2 per 50 g. Write an inequality to find 𝑀 , the weight of the chocolate you must buy with a gift box, if you are to receive a free chocolate drink.

  • A 1 0 0 𝑀 + 3 > 1 5
  • B 𝑀 2 5 + 3 β‰₯ 1 5
  • C 5 0 𝑀 + 3 > 1 5
  • D 𝑀 2 5 + 3 > 1 5
  • E 2 𝑀 + 3 β‰₯ 1 5