Inequalities Context

Kate is selling bracelets and earrings on Etsy to make money for summer vacation. The bracelets (x) cost $2 and earrings (y) cost $3. She needs to make at least $100, but can only make up to 40 pieces of jewelry. Which inequalities represent this situation?

• 2x + 3y > 100

  x + y ≤ 40

• 100x + 2y > 3

  x + y ≤ 40

• 100x + 40y > 100

  3x + 2y ≤ 40

• 3x + 2y > 100

  x + y ≤ 40

Answer explanation

The correct inequalities are 2x + 3y > 100 (total earnings) and x + y ≤ 40 (total pieces). This captures Kate's need to earn at least $100 while limiting her production to 40 items.

Cole is doing a basketball challenge for charity. In order to win money for the charity, he needs to score at least 20 points in less than 10 shots. If x is the number of 2-point shots, and y is the number of three point shots, which inequalities below represent the situation?

• x + y < 10

  2x + 3y ≥ 20

• x + y ≤ 10

  2x + 3y ≥ 20

• x + y ≤ 10

  2x + 3y > 20

• x + y < 10

  2x + 3y > 20

Answer explanation

Cole needs to score at least 20 points (2x + 3y ≥ 20) and take fewer than 10 shots (x + y < 10). Thus, the correct inequalities are x + y < 10 and 2x + 3y ≥ 20.

Write the system of inequalities
Carlos works at a movie theater selling tickets. The theater has 300 seats and charges $7.50 for adults and $5.50 for children. The theater expects to make at least $2000 for each showing

• x+y≤300
  7.5x+5.5y≥2000

• x+y<300
  7.5x+5.5y≥2000

• x+y≤300
  7.5x+5.5y≤2000

• x+y>2000
  7.5x+5.5y≤300

Answer explanation

The correct system of inequalities is x+y≤300, which ensures the number of tickets sold does not exceed the theater's capacity, and 7.5x+5.5y≥2000, which guarantees the revenue meets the minimum expectation.

JoAnn likes her job as a baby-sitter, but it pays only $3 per hour. She has been offered a job as a tutor that pays $6 per hour. Because of school, her parents only allow her to work a maximum of 15 hours per week. How many hours can JoAnn tutor and baby-sit and still make at least $66 per week? How much does she make?

• b + t < 15
  3b + 6t > 66

• b + t > 15
  3b + 6t < 66

• 3b + 6t > 15
  b + t < 66

• 3b + 6t < 15
  b + t < 66

Answer explanation

JoAnn can work a maximum of 15 hours (b + t < 15) and needs to earn at least $66 (3b + 6t > 66). The correct choice reflects these conditions, ensuring she balances her hours and earnings effectively.

Cullin needs to save at least $100 for homecoming.  He works 2 jobs earning $8/hour at Culvers and $25/hour mowing lawns.  He only has time to work 14 hours before homecoming. 

• x + y  ≥ 25
  x + y ≤ 8

• 100x + 25 y ≤ 100
  2x + 8y ≤ 14

• x + y ≥ 100
  8x + 25y ≤ 14

• 8x + 25y ≥ 100
  x + y ≤ 14

Answer explanation

Cullin needs to earn at least $100, which translates to the inequality 8x + 25y ≥ 100, where x is hours at Culvers and y is hours mowing. He can work a maximum of 14 hours, represented by x + y ≤ 14. Thus, the correct choice is 8x + 25y ≥ 100 and x + y ≤ 14.

Is (5, 2) a solution to the system?
2x+4y<20
3x-2y>11

• Yes

• No

• Sometimes

Answer explanation

To check if (5, 2) is a solution, substitute x=5 and y=2 into both inequalities. For 2x+4y<20: 2(5)+4(2)=10+8=18<20 (true). For 3x-2y>11: 3(5)-2(2)=15-4=11 (not > 11, false). Thus, the answer is No.

Write the inequality

• y>-5x-3

• y≥-5x-3

• y<-5x-3

• y≤-5x-3

Answer explanation

The correct inequality is y≥-5x-3 because it indicates that y is greater than or equal to the line represented by -5x-3, allowing for points on the line itself, which is consistent with the 'greater than or equal to' condition.

Write the inequalities

• x≤-2
  y≤x-3

• y≤-2
  y≤x-3

• x≥-2
  y≤x-3

• y≥-2
  y≤x-3

Answer explanation

The correct inequalities are y≥-2 and y≤x-3. This indicates that y is at least -2 and also less than or equal to the line defined by y=x-3, which captures the correct relationship between x and y.

The dance committee hired a DJ for the fall dance. The DJ charges $125 per hour in addition to $55 for an assistant. The committee wants to keep the total cost under $600. Which inequality represents the maximum amount of hours the DJ will play at the dance?

• 125x + 55 ≤ 600

• 125x + 55 ≥ 600

• 125x + 55 > 600

• 125x + 55 < 600

Answer explanation

The DJ charges $125 per hour plus $55 for the assistant. To keep costs under $600, the inequality is 125x + 55 < 600, where x is the number of hours. This ensures the total cost does not exceed the budget.

Which point is a solution?

• (0, 6)

• (-3, 4)

• (1, 0)

• (-4, 3)

Answer explanation

To determine which point is a solution, we check if it satisfies the equation. The point (1, 0) meets the criteria, while the others do not. Thus, (1, 0) is the correct solution.