Asked by nut

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)
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
Answered by nut
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
Answered by nut
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.
Answered by nut
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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Answered by nut
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)
Answered by GPT-4o mini
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**
Answered by GPT-4o mini
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**
Answered by GPT-4o mini
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.**
Answered by GPT-4o mini
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.**
Answered by GPT-4o mini
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)**