WRITING SYSTEM OF INEQUALITIES FROM WORD PROBLEMS

Problem 1 :

Jackie has two summer jobs. She works as a tutor, which pays $12 per hour, and she works as a lifeguard, which pays $9.50 per hour. She can work no more than 20 hours per week, but she wants to earn at least $220 per week.

Which of the following systems of inequalities represents this situation in terms of x and y, where x is the number of hours she tutors and y is the number of hours she works as a lifeguard?

Solution :

Let x be the number of hours she is working as tutor. For each hour she will be paid $12. So, the amount gained here is 12x

Let y be the number of hours as a lifeguard. For each hour she will be paid $9.50. So, the amount gained here is 9.5y

She can work no more than 20 hours means, the total number of hours she is working in both will be lesser than 20.

She wants to earn at least $220 per week, which means more than 220 is also acceptable.

12x + 9.5y ≥ 220

x + y ≤ 20

So, option C is correct.

Problem 2 :

A laundry service is buying detergent and fabric softener from its supplier. The supplier will deliver no more than 300 pounds in a shipment. Each container of detergent weighs 7.35 pounds, and each container of fabric softener weighs 6.2 pounds. The service wants to buy at least twice as many containers of detergent as containers of fabric softener.

Let d represent the number of containers of detergent, and let s represent the number of containers of fabric softener, where d and s are nonnegative integers. Which of the following systems of inequalities best represents this situation?

Solution :

Here d represents the number of containers of detergent and s represents the number of containers of fabric softener.

Weight of each container of detergent = 7.35 pounds

Weight of each container of fabric softener = 6.2 pounds

No more than 300 means, lesser than 300 is acceptable.

7.35d + 6.2s ≤ 300

At least means minimum.

The service wants to buy at least twice as many containers of detergent as containers of fabric softener.

d ≥ 2s

The system of inequalities representing this is 

7.35d + 6.2s ≤ 300

d ≥ 2s

So, option A is correct.

Problem 3 :

Roberto is an insurance agent who sells two types of policies: a $50,000 policy and a $100,000 policy. Last month, his goal was to sell at least 57 insurance policies. While he did not meet his goal, the total value of the policies he sold was over $3,000,000.

Which of the following systems of inequalities describes x, the possible number of $50,000 policies, and y, the possible number of $100,000 policies, that Roberto sold last month?

A) x + y < 57

50,000x + 100,000y < 3,000,000

B) x + y > 57

50,000x + 100,000y > 3,000,000

C) x + y < 57

50,000x + 100,000y > 3,000,000

D) x + y > 57

50,000x + 100,000y < 3,000,000

Solution :

At least 57 insurance policies, which means more than 57 is also acceptable.

x + y > 57

Since he did not meet his goal of amount more than $3,000,000.

50,000x + 100,000y < 3,000,000

The system of inequalities representing the situation is 

x + y > 57

50,000x + 100,000y < 3,000,000

So, option D is correct.

Problem 4 :

A record collector is looking to buy records that costs either $20 or $35 each. Let a be the number $20 records and b be the number of $35 records.

The collector can buy maximum of 25 records and can spend up to $750. Which of the following systems of inequalities accurately describes this relationship ?

Solution :

a = number $20 records and b = number of $35 records.

Maximum number of records he can buy is 25 and maximum amount spent will be 750.

a + b ≤ 25

20a + 35b ≤ 750

So, option D is correct.

Problem 5 :

A clothing store owner is buying T  shirts and pairs of jeans from a wholesaler. She wants to buy at least three times as many T shirts as pairs of jeans . She will spend no more than $200 on the order. Each T shirt that is ordered costs the store $1.25 and each pair of jeans that is ordered costs the store $9.75. 

Let t represents the number of T shirts ordered and j represent the number of pairs of jeans ordered. Where t and j are non negative integers. Which of the following systems of inequalities best expresses this situation ?

Solution :

Here t = number of T Shirts

j - number of pairs of jeans.

She wants to buy at least three times as many T shirts as pairs of jeans

Representing as equation :

t = 3j

At least, so we use the inequality sign ≥

t  ≥ 3j

No more than 200 means, the maximum value is 200.

1.25t + 9.75j ≤ 200

So, option C is correct.

Problem 6 :

Richard either walks or rides his bicycle everywhere he goes. When he walks, he burns 50 calories per mile, and when he rides his bicycle, he burns 25 calories per mile. Richard can travel no more than 15 miles per day, but he wants to burn at least 500 calories per day.

Which of the following systems of inequalities represents the situation in terms of w and b, where w is the number of miles he walks and b is the number of miles he rides his bicycle ?

Solution :

w is the number of miles he walks and b is the number of miles he rides his bicycle 

No more than 15 miles, then w + b ≤ 15

He burns 50 calories by walk and 25 calories by bicycle.

He wants to burn at least 500 calories per day

50w + 25b ≥ 500

So, option B is correct.