Kevin is four years older than his sister, June. The sum of their ages is 14. How old is Kevin?

Answer 1

Let Kevin's age be \( K \) and June's age be \( J \). According to the problem, we have the following two equations:

1. \( K = J + 4 \) (Kevin is four years older than June)
2. \( K + J = 14 \) (The sum of their ages is 14)

Now we can substitute the expression for \( K \) from the first equation into the second equation:

\[ (J + 4) + J = 14 \]

This simplifies to:

\[ 2J + 4 = 14 \]

Next, subtract 4 from both sides:

\[ 2J = 10 \]

Now, divide both sides by 2:

\[ J = 5 \]

Now that we know June's age, we can find Kevin's age using the first equation:

\[ K = J + 4 = 5 + 4 = 9 \]

Thus, Kevin is \( \boxed{9} \) years old.

Answer 2

Is (0,0)
a viable solution for the following system? Write either 1 for viable or 2 for nonviable as your response.

2y≥−3x−2


2y<−x+2

Answer 3

To determine if \( (0, 0) \) is a viable solution for the system of inequalities, we need to substitute \( x = 0 \) and \( y = 0 \) into both inequalities.

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

2. For the second inequality: \[ 2y < -x + 2 \] Substituting \( x = 0 \) and \( y = 0 \): \[ 2(0) < -0 + 2 \implies 0 < 2 \] This statement is also true.

Since both inequalities are satisfied by the point \( (0, 0) \), it is a viable solution.

Thus, the answer is \( \boxed{1} \) for viable.

Answer 4

A vendor sells fruit at a farmer’s market. In one day, they sold a total of 175 oranges and pears. They sold oranges for $1.50 each and pears for $2 each, making a total of $305 dollars. If x represents the number of oranges sold and y represents the number of pears sold, is (90,85)
a viable solution to the system? Write either 1 for viable or 2 for nonviable as your response.

Answer 5

To determine if \( (90, 85) \) is a viable solution for the given system, we can substitute \( x = 90 \) (the number of oranges) and \( y = 85 \) (the number of pears) into the equations based on the information provided:

1. Total number of fruits (oranges + pears): \[ x + y = 175 \]

2. Total sales from oranges and pears: \[ 1.50x + 2y = 305 \]

Now, let's substitute \( x = 90 \) and \( y = 85 \) into the first equation:

\[ 90 + 85 = 175 \] \[ 175 = 175 \quad \text{(True)} \]

Now, let's substitute \( x = 90 \) and \( y = 85 \) into the second equation:

\[ 1.50(90) + 2(85) = 305 \] Calculating each term: \[ 135 + 170 = 305 \] \[ 305 = 305 \quad \text{(True)} \]

Both equations are satisfied by the point \( (90, 85) \).

Thus, \( (90, 85) \) is a viable solution.

The answer is \( \boxed{1} \) for viable.

Answer 6

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≥n+2

0.25p+4n≤100

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

0.25p+4n≥100

Answer 7

To represent the constraints of Amy's scenario as a system of inequalities where \( p \) is the number of pencils and \( n \) is the number of notebooks, we need to examine the two conditions described:

1. The number of pencils needed is at least twice as many as the number of notebooks. This can be represented by the inequality: \[ p \geq 2n \]

2. The total cost of the pencils and notebooks should not exceed her budget of $100. The cost of pencils is $0.25 each and notebooks is $4 each. This gives the inequality: \[ 0.25p + 4n \leq 100 \]

Now we can combine these two inequalities into a system of inequalities:

\[ \begin{cases} p \geq 2n \ 0.25p + 4n \leq 100 \end{cases} \]

From the choices given, the correct representation of the constraints is:

Option:
p ≥ 2n
0.25p + 4n ≤ 100

Thus, the answer is:
p greater-than-or-equal-to 2 n
0.25 p plus 4 n less-than-or-equal-to 100
So the selection corresponds to the second option you provided.

Answer 8

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-or-equal-to 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 25 s less-than-or-eqaul-to 375

w≥25

s<375

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

Answer 9

To represent Adam's financial situation as a system of linear inequalities, we need to consider the constraints given in the problem:

1. Adam has at least $25 in his wallet. This means the amount in his wallet \( w \) must be greater than or equal to 25, which can be represented as: \[ w \geq 25 \]

2. Adam has no more than $375 in his savings. This means the amount in his savings \( s \) must be less than or equal to 375, which can be represented as: \[ s \leq 375 \]

Putting these two inequalities together, we have the system:

\[ \begin{cases} w \geq 25 \ s \leq 375 \end{cases} \]

From the options provided, the correct representation is:

Option:
w ≥ 25
s ≤ 375

This corresponds to the response:
w greater-than-or-equal-to 25 s less-than-or-equal-to 375.

Thus, the final answer is: w greater-than-or-equal-to 25 s less-than-or-equal-to 375.

Answer 10

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

y<−5x+2

3y>3x−6

(1 point)
Responses

(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.
left parenthesis 0 comma negative 3 right parenthesis is a viable solution since it makes one inequality 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 nonviable solution since it does not make both inequalities true.

Answer 11

To determine if \( (0, -3) \) is a viable solution for the given system of inequalities, we will substitute \( x = 0 \) and \( y = -3 \) into both inequalities:

1. First inequality: \[ y < -5x + 2 \] Substituting \( x = 0 \) and \( y = -3 \): \[ -3 < -5(0) + 2 \] This simplifies to: \[ -3 < 2 \quad \text{(True)} \]

2. Second inequality: \[ 3y > 3x - 6 \] Substituting \( x = 0 \) and \( y = -3 \): \[ 3(-3) > 3(0) - 6 \] This simplifies to: \[ -9 > -6 \quad \text{(False)} \]

Now, since the first inequality is true and the second inequality is false, \( (0, -3) \) does not satisfy both inequalities.

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

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