Amy needs to purchase pencils and notebooks for her students. The number of pencils needed is at least twice as many as the number of notebooks. The pencils cost $0.25 each and the notebooks cost $4. Amy’s budget for these supplies allows for a maximum cost of $100. How do you represent the constraints of this scenario as a system of inequalities, where p is the number of pencils and n is the number of notebooks Amy will buy?(1 point)

Question

Responses

p≥2n

0.25p+4n≥100

p greater-than-or-equal-to 2 n 0.25 p plus 4 n greater-than-or-equal-to 100

p≥2n

0.25p+4n≤100

p greater-than-or-equal-to 2 n 0.25 p plus 4 n less-than-or-equal-to 100

p<2n

0.25p+4n≤100

p less than 2 n 0.25 p plus 4 n less-than-or-equal-to 100

p≥n+2

0.25p+4n≤100

Answer 1

To represent the constraints of Amy's situation as a system of inequalities, we need to consider the two main conditions given:

1. The number of pencils needed (p) is at least twice the number of notebooks (n), which can be represented by the inequality: \[ p \geq 2n \]

2. The total cost of the pencils and notebooks must not exceed Amy’s budget of $100. The cost of pencils is $0.25 each and the cost of notebooks is $4 each. Therefore, this constraint can be represented by the inequality: \[ 0.25p + 4n \leq 100 \]

Considering these two inequalities together, the correct representation of the constraints is: \[ p \geq 2n \] \[ 0.25p + 4n \leq 100 \]

So the final response that accurately reflects this is: p≥2n
0.25p+4n≤100

Answer 2

Adam has at least $25 in his wallet and no more than $375 in his savings. How can Adam represent this scenario as a system of linear inequalities, where w is the amount of money in his wallet and s is the amount of money in his savings?(1 point)
Responses

w>25

s≤375

w greater than 25 s less-than-or-eqaul-to 375

w≥25

s>375

w greater-than-or-equal-to 25 s greater than 375

w≥25

s<375

w greater-than-or-equal-to 25 s less than 375

w≥25

s≤375

Answer 3

To represent Adam's financial situation as a system of linear inequalities, we can use the following variables:

• \( w \) represents the amount of money in Adam's wallet.
• \( s \) represents the amount of money in his savings.

Given the conditions:

1. Adam has at least $25 in his wallet, which can be represented as: \[ w \geq 25 \]

2. Adam has no more than $375 in his savings, which can be represented as: \[ s \leq 375 \]

So, the system of inequalities that represents this scenario is: \[ w \geq 25 \] \[ s \leq 375 \]

The correct response that captures this representation is: w≥25 s≤375

Answer 4

Is (0,−3) a viable solution for the following system?

y<−5x+2

3y>3x−6

(1 point)
Responses

(0,−3) is a nonviable solution since it does not make both inequalities true.
left parenthesis 0 comma negative 3 right parenthesis is a nonviable solution since it does not make both inequalities true.

(0,−3) is a nonviable solution since it makes both inequalities true.
left parenthesis 0 comma negative 3 right parenthesis is a nonviable solution since it makes both inequalities true.

(0,−3) is a viable solution since it makes both inequalities true.
left parenthesis 0 comma negative 3 right parenthesis is a viable solution since it makes both inequalities true.

(0,−3) is a viable solution since it makes one inequality true.

Answer 5

To determine whether \((0, -3)\) is a viable solution for the system of inequalities given, we need to evaluate it in both inequalities:

1. For the first inequality: \[ y < -5x + 2 \] Substituting \(x = 0\) and \(y = -3\): \[ -3 < -5(0) + 2 \implies -3 < 2 \] This inequality is true.

2. For the second inequality: \[ 3y > 3x - 6 \] Again substituting \(x = 0\) and \(y = -3\): \[ 3(-3) > 3(0) - 6 \implies -9 > -6 \] This inequality is false.

Since \((0, -3)\) satisfies the first inequality but does not satisfy the second inequality, it cannot be considered a viable solution for the system of inequalities as both inequalities must hold true.

Therefore, the correct response is: (0,−3) is a nonviable solution since it does not make both inequalities true.

Answer 6

Is (−2,2) a viable solution for the following system? Explain.

−7x+2y=18

3x+3y=0

(1 point)
Responses

(−2,2) is a viable solution since it makes both equations true.
left parenthesis negative 2 comma 2 right parenthesis is a viable solution since it makes both equations true.

(−2,2) is a nonviable solution since it makes neither equation true.
left parenthesis negative 2 comma 2 right parenthesis is a nonviable solution since it makes neither equation true.

(−2,2) is a nonviable solution since it makes one equation true.
left parenthesis negative 2 comma 2 right parenthesis is a nonviable solution since it makes one equation true.

(−2,2) is a viable solution since it makes one equation true.
left parenthesis negative 2 comma 2 right parenthesis is a viable solution since it makes one equation true.
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Answer 7

To determine whether \((-2, 2)\) is a viable solution for the system of equations, we need to substitute \(x = -2\) and \(y = 2\) into both equations:

1. For the first equation: \[ -7x + 2y = 18 \] Substituting \(x = -2\) and \(y = 2\): \[ -7(-2) + 2(2) = 18 \ 14 + 4 = 18 \ 18 = 18 \] This equation is true.

2. For the second equation: \[ 3x + 3y = 0 \] Substituting \(x = -2\) and \(y = 2\): \[ 3(-2) + 3(2) = 0 \ -6 + 6 = 0 \ 0 = 0 \] This equation is also true.

Since \((-2, 2)\) satisfies both equations, it is a viable solution.

The correct response is: (−2,2) is a viable solution since it makes both equations true.

Answer 8

The sum of twice the number of green marbles and the number of yellow marbles in a bag is more than 20. The total number of green and yellow marbles is less than 15. Which of the following is a viable solution to the system of inequalities, where x is the number of green marbles and y is the number of yellow marbles?(1 point)
Responses

(6,7)
left parenthesis 6 comma 7 right parenthesis

(10,5)
left parenthesis 10 comma 5 right parenthesis

(3,12)
left parenthesis 3 comma 12 right parenthesis

(8,6)

Answer 9

To determine which pair of values is a viable solution to the system of inequalities based on the given conditions, we first need to translate the problem into inequalities:

1. The sum of twice the number of green marbles and the number of yellow marbles is more than 20: \[ 2x + y > 20 \]

2. The total number of green and yellow marbles is less than 15: \[ x + y < 15 \]

Next, we will test each option provided to see if it satisfies both inequalities.

1. For (6, 7):

   • \(2x + y = 2(6) + 7 = 12 + 7 = 19\) (not > 20)
   • \(x + y = 6 + 7 = 13\) (< 15)
   • Result: Does not satisfy the first inequality.

2. For (10, 5):

   • \(2x + y = 2(10) + 5 = 20 + 5 = 25\) (> 20)
   • \(x + y = 10 + 5 = 15\) (not < 15)
   • Result: Does not satisfy the second inequality.

3. For (3, 12):

   • \(2x + y = 2(3) + 12 = 6 + 12 = 18\) (not > 20)
   • \(x + y = 3 + 12 = 15\) (not < 15)
   • Result: Does not satisfy either inequality.

4. For (8, 6):

   • \(2x + y = 2(8) + 6 = 16 + 6 = 22\) (> 20)
   • \(x + y = 8 + 6 = 14\) (< 15)
   • Result: Satisfies both inequalities.

Since (8, 6) satisfies both inequalities, it is the viable solution.

Thus, the correct response is: (8, 6)