Worksheet: Applications on Systems of Inequalities

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 π‘₯ and its width by 𝑦.

  • Aπ‘₯>88, π‘₯+𝑦>253
  • Bπ‘₯>88, π‘₯+𝑦<253
  • Cπ‘₯>88, 2(π‘₯+𝑦)<253
  • Dπ‘₯β‰₯88, 2(π‘₯+𝑦)<253
  • Eπ‘₯<88, 2(π‘₯+𝑦)<253

Q2:

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 π‘₯ to represent the number of questions answered from section A and 𝑦 to represent the number from section B.

  • Aπ‘₯≀4, 𝑦≀6, π‘₯+𝑦≀11, 3π‘₯+6𝑦=100
  • Bπ‘₯>4, 𝑦β‰₯6, π‘₯+𝑦β‰₯11, 3π‘₯+6𝑦β‰₯100
  • Cπ‘₯β‰₯4, 𝑦β‰₯6, π‘₯+𝑦β‰₯11, 3π‘₯+6𝑦β‰₯100
  • Dπ‘₯>4, 𝑦>6, π‘₯+𝑦β‰₯11, 3π‘₯+6𝑦≀100
  • Eπ‘₯β‰₯4, 𝑦β‰₯6, π‘₯+𝑦β‰₯11, 3π‘₯+6𝑦≀100

Q3:

Anthony and Daniel went on a tour driving to Luxor and Aswan and took turns driving; each day Anthony drove for at least 4 hours and NO more than 8 hours, while Daniel drove at least 2 hours and less than 7 hours. The total time they drove daily was NO more than 9 hours. State the system of inequalities that describes the situation, using π‘₯ to represent the number of hours that Anthony drove and 𝑦 to represent the number of hours that Daniel drove.

  • A4≀π‘₯<8, 2≀𝑦<7, π‘₯+𝑦≀9
  • B4≀π‘₯≀8, 2≀𝑦<7, π‘₯+𝑦≀9
  • C4≀π‘₯≀8, 2≀𝑦≀7, π‘₯+𝑦≀9
  • D4≀π‘₯≀8, 2≀𝑦<7, π‘₯+𝑦>9
  • E4≀π‘₯≀8, 2≀𝑦≀7, π‘₯+𝑦<9

Q4:

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 π‘₯ to represent the amount of the first type and 𝑦 to represent the second type, state the system of inequalities that represents this situation.

  • Aπ‘₯β‰₯6, 𝑦β‰₯9, 5π‘₯+7𝑦<55
  • Bπ‘₯>5, 𝑦>7, 6π‘₯+9𝑦<55
  • Cπ‘₯β‰₯5, 𝑦β‰₯7, 6π‘₯+9𝑦≀55
  • Dπ‘₯β‰₯5, 𝑦β‰₯7, 6π‘₯+9𝑦<55
  • Eπ‘₯β‰₯6, 𝑦β‰₯9, 5π‘₯+7𝑦≀55

Q5:

During a trip to the zoo you decide to buy peanuts and berries. A pack of peanuts costs 73pounds and a pack of berries costs 52pounds. Given that you want to spend not more than126 pounds, write an inequality describing how many packs of each you could buy.

  • A73π‘₯+52𝑦β‰₯126
  • B73π‘₯+52𝑦≀126
  • C73π‘₯+52𝑦>126
  • D73π‘₯+52𝑦<126

Q6:

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

  • Aπ‘₯β‰₯0, 𝑦β‰₯0, 𝑦<61, 2(π‘₯+𝑦)<177
  • Bπ‘₯β‰₯0, 𝑦β‰₯0, 𝑦β‰₯61, 2(π‘₯+𝑦)<177
  • Cπ‘₯β‰₯0, 𝑦β‰₯0, 𝑦>61, 2(π‘₯+𝑦)<177
  • Dπ‘₯β‰₯0, 𝑦β‰₯0, 𝑦>61, π‘₯+𝑦<177
  • Eπ‘₯β‰₯0, 𝑦β‰₯0, 𝑦β‰₯61, π‘₯+𝑦>177

Q7:

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

  • Option A: a base salary of $14,000 a year with a commission of 10% of your sales
  • Option B: a base salary of $19,000 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 $83,333.33
  • BMore than $35,714.29
  • CMore than $235,714.29
  • DMore than $550,000

Q8:

While on a trip you decide to buy cashews and pistachios. Given that you want to spend less than 204 LE, the figure below illustrates the relation between the number of kilograms of cashews and pistachios you can buy. Determine the price of a kilogram of cashews and a kilogram of pistachios.

  • Acashews = 68 LE, pistachios = 51 LE
  • Bcashews = 136 LE, pistachios = 153 LE
  • Ccashews = 51 LE, pistachios = 68 LE
  • Dcashews = 3 LE, pistachios = 4 LE
  • Ecashews = 4 LE, pistachios = 3 LE

Q9:

James 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 π‘₯ to represent the number of small candles and 𝑦 to represent the number of large candles.

  • Aπ‘₯+𝑦≀20, 3π‘₯+5𝑦β‰₯80
  • Bπ‘₯β‰₯0, 𝑦β‰₯0, π‘₯+𝑦β‰₯20, 3π‘₯+5𝑦≀80
  • Cπ‘₯+𝑦β‰₯20, 3π‘₯+5𝑦β‰₯80
  • Dπ‘₯β‰₯0, 𝑦β‰₯0, π‘₯+𝑦≀20, 3π‘₯+5𝑦≀80
  • Eπ‘₯β‰₯0, 𝑦β‰₯0, π‘₯+𝑦β‰₯20, 3π‘₯+5𝑦β‰₯80

Q10:

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 π‘₯ 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).

  • Aπ‘₯β‰₯0, 𝑦β‰₯0, 2π‘₯+3𝑦≀120, 3π‘₯+2𝑦β‰₯100
  • B2π‘₯+3𝑦β‰₯120, 3π‘₯+2𝑦≀100
  • Cπ‘₯β‰₯0, 𝑦β‰₯0, 2π‘₯+3𝑦β‰₯100, 3π‘₯+2𝑦β‰₯120
  • D2π‘₯+3𝑦β‰₯100, 3π‘₯+2𝑦β‰₯120
  • Eπ‘₯β‰₯0, 𝑦β‰₯0, 2π‘₯+3𝑦≀100, 3π‘₯+2𝑦≀120

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