Lesson Worksheet: Applications on Systems of Inequalities Mathematics

In this worksheet, we will practice solving applications of systems of inequalities by translating each condition into an inequality.

Q1:

A shepherd wants to build a rectangular sheep barn. The length of the barn must be more than 88 m and its perimeter must be less than 253 m. Derive the system of inequalities that describes the situation, denoting the length of the barn by 𝑥 and its width by 𝑦.

  • A𝑥>88, 𝑦>0, 𝑥+𝑦>253
  • B𝑥>88, 𝑦<0, 𝑥+𝑦<253
  • C𝑥>88, 𝑦>0, 2(𝑥+𝑦)<253
  • D𝑥88, 𝑦>0, 2(𝑥+𝑦)<253
  • E𝑥<88, 𝑦<0, 2(𝑥+𝑦)<253

Q2:

A carpenter wants to buy two types of nails; the first type costs 6 pounds per kilogram, and the second type costs 9 pounds per kilogram. He needs at least 5 kg of the first type and at least 7 kg of the second. He can spend less than 55 pounds. Using 𝑥 to represent the amount of the first type and 𝑦 to represent the second type, state the system of inequalities that represents this situation.

  • A𝑥6, 𝑦9, 5𝑥+7𝑦<55
  • B𝑥>5, 𝑦>7, 6𝑥+9𝑦<55
  • C𝑥5, 𝑦7, 6𝑥+9𝑦55
  • D𝑥5, 𝑦7, 6𝑥+9𝑦<55
  • E𝑥6, 𝑦9, 5𝑥+7𝑦55

Q3:

Charlotte wants to make dresses and suits. Each dress or suit will have the same quantity of cloth and the same number of buttons.

The following inequality represents the number of dresses (𝐷) and the number of suits (𝑆) that she can make with 25 m2 of cloth: 5𝐷+7𝑆<25.

Additionally, the following inequality represents the number of dresses (𝐷) and the number of suits (𝑆) that she can make with 100 buttons: 12𝐷+18𝑆<100.

Given that she has 25 m2 of cloth and 100 buttons, does she have enough cloth to make 2 dresses and 3 suits?

  • AYes
  • BNo

Q4:

A shepherd wants to build a rectangular sheep barn, and the graph represents the relation between the dimensions of the needed barn, where 𝑥 represents the width and 𝑦 represents the length. State the system of inequalities that describes the dimensions of the barn.

  • A𝑥0, 𝑦0, 𝑦<61, 2(𝑥+𝑦)<177
  • B𝑥0, 𝑦0, 𝑦61, 2(𝑥+𝑦)<177
  • C𝑥0, 𝑦0, 𝑦>61, 2(𝑥+𝑦)<177
  • D𝑥0, 𝑦0, 𝑦>61, 𝑥+𝑦<177
  • E𝑥0, 𝑦0, 𝑦61, 𝑥+𝑦>177

Q5:

A company manufactures a product in two different production plants, 𝑃 and 𝑃. 𝑃 produces 20 units per month and 𝑃 produces 60 units per month. Each month, the company supplies at least 40 units of this product to customer 1 and at least 8 units to customer 2.

Customer 1 is supplied with 20% and 80% of their needs from 𝑃 and 𝑃, respectively, while customer 2 is supplied with 40% and 60% of their needs from 𝑃 and 𝑃 respectively.

Which part of the following graph represents the number of units supplied to both customers each month?

  • AA, C, E, and F
  • BD
  • CF
  • DB and D
  • EH

Q6:

A toy factory produces two types of planes, 2-engine planes and 4-engine planes. Each 2-engine plane requires 6 hours in the assembly department and one hour in the quality control department, and each 4-engine plane requires 8 hours in the assembly department and 2 hours in the quality control department.

The maximum number of working hours per week is 120 in the assembly department and 25 in the quality control department.

Which of the following is the graph representing the number of planes produced per week?

  • A
  • B
  • C
  • D
  • E

Q7:

A baby food factory produces two types of baby food. The first type contains 2 units of vitamin (A) and 3 units of vitamin (B) per gram. The second type contains 3 units of vitamin (A) and 2 units of vitamin (B) per gram. If a baby needs at least 100 units of vitamin (A) and 120 units of vitamin (B) per day, state the system of inequalities that describes the food that the baby must eat each day to meet these requirements. Use 𝑥 to represent the mass of the first type of baby food (in grams) and 𝑦 to represent the mass of the second type of baby food (in grams).

  • A𝑥0, 𝑦0, 2𝑥+3𝑦120, 3𝑥+2𝑦100
  • B2𝑥+3𝑦120, 3𝑥+2𝑦100
  • C𝑥0, 𝑦0, 2𝑥+3𝑦100, 3𝑥+2𝑦120
  • D2𝑥+3𝑦100, 3𝑥+2𝑦120
  • E𝑥0, 𝑦0, 2𝑥+3𝑦100, 3𝑥+2𝑦120

Q8:

A candy manufacturer has 30 kg of chocolate cookies and 60 kg of vanilla cookies. Sales will be made in two different combinations. The first combination will be one-quarter chocolate cookies and three-quarters vanilla cookies by weight, while the second combination will be half chocolate cookies and half vanilla cookies by weight. There is a contract requiring that at least 20 kg of the second combination should be supplied to a specific bakery.

Which of the following systems of inequalities represents the number of kilograms of the first and second combinations that will be sold?

Let 𝑥 be the number of kilograms of the first combination and 𝑦 the number of kilograms of the second combination.

  • A𝑥+2𝑦120,
    3𝑥+2𝑦240,
    𝑦20,
    𝑥0
  • B𝑥+2𝑦120,
    3𝑥+2𝑦240,
    𝑦20,
    𝑥0,
    𝑦0
  • C𝑥+2𝑦120,
    3𝑥+2𝑦240,
    𝑦20
  • D𝑥+2𝑦120,
    3𝑥+2𝑦240,
    𝑥20
  • E𝑥+2𝑦120,
    3𝑥+2𝑦240,
    𝑦20,
    𝑥0

Q9:

David is going to the store to buy candles. Small candles cost $3 and large candles cost $5. He needs to buy at least 20 candles, and he cannot spend more than $80. Write a system of linear inequalities that represents the situation, using 𝑥 to represent the number of small candles and 𝑦 to represent the number of large candles.

  • A𝑥+𝑦20, 3𝑥+5𝑦80
  • B𝑥0, 𝑦0, 𝑥+𝑦20, 3𝑥+5𝑦80
  • C𝑥+𝑦20, 3𝑥+5𝑦80
  • D𝑥0, 𝑦0, 𝑥+𝑦20, 3𝑥+5𝑦80
  • E𝑥0, 𝑦0, 𝑥+𝑦20, 3𝑥+5𝑦80

Q10:

A teacher gave his students 100 minutes to solve a test that has two sections: section A and section B. The students had to answer at least 4 questions from section A and at least 6 questions from section B and answer at least 11 questions in total. If a girl answered each question in section A in 3 minutes and each question in section B in 6 minutes, derive the system of inequalities that would help to know how many questions she tried to solve in each section. Use 𝑥 to represent the number of questions answered from section A and 𝑦 to represent the number from section B.

  • A𝑥4, 𝑦6, 𝑥+𝑦11, 3𝑥+6𝑦=100
  • B𝑥>4, 𝑦6, 𝑥+𝑦11, 3𝑥+6𝑦100
  • C𝑥4, 𝑦6, 𝑥+𝑦11, 3𝑥+6𝑦100
  • D𝑥>4, 𝑦>6, 𝑥+𝑦11, 3𝑥+6𝑦100
  • E𝑥4, 𝑦6, 𝑥+𝑦11, 3𝑥+6𝑦100

This lesson includes 22 additional questions and 288 additional question variations for subscribers.

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