FINDING RANGE OF VALUES INEQUALITY PROBLEMS

Problem 1 :

Find the range of values of x that satisfies both

3(x + 2) ≤ 30 and 4x + 3 > 21

Solution :

Solving 3(x + 2) ≤ 30

Dividing by 3 on both sides

x + 2 ≤ 30/3

x + 2 ≤ 10

Subtracting 2 on both sides

x ≤ 8

Solving 4x + 3 > 21

4x > 21 - 3

4x > 18

x > 18/4

x > 4.5

Combing the values of x, we get

(4.5, 8]

Problem 2 :

Liam brought $28 to the arcade and played games that cost 75 cents each. Laney brought $12 to the arcade and spent $4.25 on a slice of pizza and a Coke. How many games can Liam play so that he leaves with less money than Laney?

Solution :

Let x be the number of games that Liam played.

Amount spent on games = 0.75x

Amount she has remaining = (28 - 0.75 x)

Amount spent for pizza and coke = $4.25

Amount that Laney has remaining = 12 - 4.25

The amount that Liam has to be less than Laney

28 - 0.75x < 12 - 4.25

28 - 0.75x < 7.75

-0.75x < 7.75 - 28

- 0.75x < -20.25

Dividing by - on both sides

x > 20.25/0.75

x > 27

So, Liam can play more than 27 games and maximum 37 games.

Problem 3 :

Christina goes to the market with $50. She buys some papayas for $20 and spends the rest of the money on bananas. Each banana cost $0.60. Write an inequality for the number of bananas she can purchase and solve.

Solution :

Amount Christina has = $50

Cost of papayas = $20

Cost of each banana = $0.60

Number of bananas she is purchasing = x

20 + 0.60x ≤ 50

Solving for x,

0.60x ≤ 50 - 20

0.60x ≤ 30

x ≤ 30/0.60

x ≤ 50

So, he can buy 50 or lesser than 50 bananas.

Problem 4 :

Billy goes to the store. He has $90. He wants to purchase a leather jacket for $45, a hat for $10, and the rest on jeans. Each pair of jeans costs $35. Write an inequality for the number of jeans he can purchase and solve.

Solution :

Amount he has = $90

Cost of leather jacket = $45

Cost of hat = $10

Let x be the number of pair of jeans.

Cost of pair of jeans = $35

45 + 10 + 35x ≤ 90

Solving for x,

55 + 35x ≤ 90

35x ≤ 90 - 55

35x ≤ 35

x ≤ 35/35

x ≤ 1

So, he can buy maximum one pair of jeans.

Problem 5 :

Adrian works in New York City and makes $42 per hour. She works in an office and must get her suit dry cleaned every day for $75. If she wants to end up with more than $260 a day, at least how many hours must she work?

Solution :

Let x be the number of hours she is working.

75 + 42x > 260

Solving for x,

42x > 260 - 75

42x > 185

x > 185/42

x > 4.40

She has to work more than 4 hours to earn more than 260.

Problem 6 :

Erin has $50. She wants to purchase a cell phone ($20) and spend the rest on music CDs. Each music CD costs $8. Write and solve an inequality for the number of music CDs she can purchase.

Solution :

Amount Erin has = $50

Cost of cell phone = $20

Cost of each CD = $8

Let x be the number of CD's

20 + 8x < 50

Subtracting 20

8x < 50 - 20

8x < 30

x < 30/8

x < 3.75

It is not possible to purchase 3.75 CDs. So, she can buy 3 or less number of CDs. Then the range is [0, 3].

Problem 7 :

Jason is saving up to buy a digital camera that costs $490. So far, he saved $175. He would like to buy the camera 3 weeks from now. Write an inequality to represent at least how much he must save every week to have enough money to purchase the camera. Solve

Solution :

Let x be the amount to be saved on each week.

175 + 3x ≥ 490

Solving for x,

3x ≥ 490 - 175

3x ≥ 315

x ≥ 315/3

x ≥ 105

So, the least amount to be saved for each week is 105.