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At the snack bar, hot dogs cost $4 each and bottled water costs $2 each. In the first business hour of the day, less than $12 worth of hot dogs and water were sold.

Which are reasonable solutions for this situation if x represents the number of hot dogs sold and y represents the number of bottles of water sold? Check all that apply.

(–1, 5)
(0, 6)
(2, 1)
(1, 1.5)
(1, 3)
(2, 2)

11 answers

To determine which solutions are reasonable given that hot dogs cost $4 each and bottled water costs $2 each, and that less than $12 worth of hot dogs and water were sold, we can set up the inequality:

\[ 4x + 2y < 12 \]

where \(x\) is the number of hot dogs sold and \(y\) is the number of bottles of water sold. We will evaluate each of the provided pairs \((x, y)\) to see if they satisfy this inequality.

  1. (–1, 5): \[ 4(-1) + 2(5) = -4 + 10 = 6 \quad \text{(valid, since } 6 < 12\text{)} \]

  2. (0, 6): \[ 4(0) + 2(6) = 0 + 12 = 12 \quad \text{(not valid, since } 12 \not< 12\text{)} \]

  3. (2, 1): \[ 4(2) + 2(1) = 8 + 2 = 10 \quad \text{(valid, since } 10 < 12\text{)} \]

  4. (1, 1.5): \[ 4(1) + 2(1.5) = 4 + 3 = 7 \quad \text{(valid, since } 7 < 12\text{)} \]

  5. (1, 3): \[ 4(1) + 2(3) = 4 + 6 = 10 \quad \text{(valid, since } 10 < 12\text{)} \]

  6. (2, 2): \[ 4(2) + 2(2) = 8 + 4 = 12 \quad \text{(not valid, since } 12 \not< 12\text{)} \]

Now, we can summarize the results:

  • Valid solutions: (–1, 5), (2, 1), (1, 1.5), (1, 3)
  • Invalid solutions: (0, 6), (2, 2)

Reasonable solutions for the situation are:

  • (–1, 5)
  • (2, 1)
  • (1, 1.5)
  • (1, 3)
(2, 1)
(1, 3)

Will likes two brands of healthy breakfast cereal. In Superfiber cereal, there are 5 grams of fiber in one cup. In Fiber Oats cereal, there are 4 grams of fiber in one cup. Let x represent the number of cups of Superfiber Will ate this week and let y represent the number cups of Fiber Oats he ate this week. Which inequality represents the situation if the cereal Will ate this week contained at least 30 grams of fiber?

5x + 4y ≥ 30
5x + 4y ≤ 30
4x + 5y ≥ 30
4x + 5y ≤ 30

To represent the situation about the amount of fiber Will consumed from two brands of cereal, we need to use the given information about fiber content in each type:

  • Superfiber cereal: 5 grams of fiber per cup
  • Fiber Oats cereal: 4 grams of fiber per cup

Let:

  • \(x\) = number of cups of Superfiber Will ate
  • \(y\) = number of cups of Fiber Oats he ate

The total amount of fiber from both cereals can be expressed as: \[ 5x + 4y \]

Since we are told that the total fiber consumed is at least 30 grams, the appropriate inequality is:

\[ 5x + 4y \geq 30 \]

Thus, the correct inequality representing the situation is:

5x + 4y ≥ 30

Felix bought x pounds of grapes that cost $1.25 per pound and y boxes of cereal that cost $2.50 per box. He spent less than $10. Which graph represents this scenario? 1. On a coordinate plane, a dashed straight line has a negative slope and goes through (0, 4) and (4, 2). Everything above and to the right of the line is shaded. 2. On a coordinate plane, a dashed straight line has a negative slope and goes through (0, 4) and (4, 2). Everything below and to the left of the line is shaded. 3. On a coordinate plane, a dashed straight line has a positive slope and goes through (negative 1, negative 4) and (1, 0). Everything to the right of the line is shaded. 4. On a coordinate plane, a dashed straight line has a positive slope and goes through (negative 1, negative 4) and (1, 0). Everything to the left of the line is shaded.

To represent Felix's situation using an inequality, we can express the total cost of grapes and cereal mathematically.

  • The cost of \(x\) pounds of grapes at $1.25 per pound is \(1.25x\).
  • The cost of \(y\) boxes of cereal at $2.50 per box is \(2.50y\).

If Felix spent less than $10, we can write the inequality as:

\[ 1.25x + 2.50y < 10 \]

To analyze this inequality, we can express it in slope-intercept form (y = mx + b) to identify the line and the shaded region:

  1. First, we isolate \(y\): \[ 2.50y < 10 - 1.25x \] \[ y < \frac{10 - 1.25x}{2.50} \] \[ y < 4 - 0.5x \]

  2. This line has a negative slope of \(-0.5\) and a y-intercept of \(4\). The line would cross the y-axis at \( (0, 4) \) and for \(x = 4\): \[ y = 4 - 0.5(4) = 4 - 2 = 2 \quad \Rightarrow \quad (4, 2) \]

This means that the line going through the points \((0, 4)\) and \((4, 2)\) is indeed the correct line.

  1. Since the inequality is \(y < 4 - 0.5x\), we need to shade below the line, indicating all points where \(1.25x + 2.50y\) is less than $10.

Analyzing the options:

  1. The first option describes shading above the line, which is incorrect.
  2. The second option describes shading below the line, which is correct.
  3. The third option describes a line with a positive slope, which does not apply here.
  4. The fourth option also describes a line with a positive slope, so it doesn't fit either.

Conclusively, the correct graph that represents this scenario is:

2. On a coordinate plane, a dashed straight line has a negative slope and goes through (0, 4) and (4, 2). Everything below and to the left of the line is shaded.

Alina spent no more than $45 on gas for a road trip. The first gas station she used charged $3.50 per gallon and the second gas station charged $4.00 per gallon. Which inequality relates the number of gallons of gas she bought at the first station, x, the number of gallons of gas she bought at the second station, y, and the total amount she paid? What are the possible values of y?

3.5x + 4y ≥ 45, y ≥ 11.25
3.5x + 4y ≥ 45, 0 ≤ y ≤ 10.375
3.5x + 4y ≤ 45, 0 ≤ y ≤ 11.25
3.5x + 4y ≤ 45, y ≥ 10.375

To determine the appropriate inequality for Alina's gas expenditures and the feasible range for the number of gallons of gas purchased at each station, let's start by breaking down the information given:

  1. Total spending limit: Alina spent no more than $45 on gas.
  2. Cost per gallon:
    • First gas station: $3.50 per gallon (represented by \(x\) gallons)
    • Second gas station: $4.00 per gallon (represented by \(y\) gallons)

The total cost for the gas from both stations can be written as:

\[ 3.50x + 4.00y \]

Considering that Alina spent no more than $45, we can represent this scenario with the inequality:

\[ 3.50x + 4y \leq 45 \]

Next, we will find the possible values for \(y\):

  • To determine the maximum value of \(y\), we can isolate \(y\) in the equation derived from the inequality:

\[ 4y \leq 45 - 3.50x \]

\[ y \leq \frac{45 - 3.50x}{4} \]

To find the maximum value of \(y\), we need to evaluate it when \(x\) is minimized (i.e., \(x = 0\)).

If Alina bought no gas from the first station (\(x = 0\)): \[ y \leq \frac{45}{4} = 11.25 \]

Thus, \(0 \leq y \leq 11.25\) since \(y\) cannot be negative (she can buy 0 or more gallons of gas).

Summary:

  1. The correct inequality that relates the gallons of gas she bought at each gas station and her total spending is: \[ 3.5x + 4y \leq 45 \]

  2. The possible values for \(y\) are: \[ 0 \leq y \leq 11.25 \]

Among the provided options, the most fitting one is:

3.5x + 4y ≤ 45, 0 ≤ y ≤ 11.25.

Penelope went to the store to buy x pounds of broccoli for $1.10 per pound and y cans of soup for $2.50 each. In total, she spent less than $10. The inequality relating the purchases she made and the total purchase price is
1.10x + 2.50y < 10. Which are reasonable solutions for this situation? Check all that apply.

(–1, 4)
(0, 2)
(3, 2.5)
(2, 4)
(0.5, 3.78)
(1.5, 3)

To find the reasonable solutions for the given inequality relating to Penelope's purchases, we first note the provided inequality:

\[ 1.10x + 2.50y < 10 \]

We need to evaluate whether each pair of values \((x, y)\) satisfies this inequality. Let's check each option:

  1. (–1, 4): \[ 1.10(-1) + 2.50(4) = -1.10 + 10 = 8.90 \quad \text{(valid, since } 8.90 < 10\text{)} \]

  2. (0, 2): \[ 1.10(0) + 2.50(2) = 0 + 5 = 5.00 \quad \text{(valid, since } 5.00 < 10\text{)} \]

  3. (3, 2.5): \[ 1.10(3) + 2.50(2.5) = 3.30 + 6.25 = 9.55 \quad \text{(valid, since } 9.55 < 10\text{)} \]

  4. (2, 4): \[ 1.10(2) + 2.50(4) = 2.20 + 10.00 = 12.20 \quad \text{(not valid, since } 12.20 \not< 10\text{)} \]

  5. (0.5, 3.78): \[ 1.10(0.5) + 2.50(3.78) = 0.55 + 9.45 = 10.00 \quad \text{(not valid, since } 10.00 \not< 10\text{)} \]

  6. (1.5, 3): \[ 1.10(1.5) + 2.50(3) = 1.65 + 7.50 = 9.15 \quad \text{(valid, since } 9.15 < 10\text{)} \]

Summary of Valid Solutions:

  • Valid solutions: (–1, 4), (0, 2), (3, 2.5), (1.5, 3)
  • Invalid solutions: (2, 4), (0.5, 3.78)

Therefore, the reasonable solutions for this situation are:

  • (–1, 4)
  • (0, 2)
  • (3, 2.5)
  • (1.5, 3)
(0,2)
(1.5,3)

Alexia spent 3 minutes working on each of her math problems and 4 minutes on each of her science problems. Her homework took her more than 60 minutes to complete. The boundary line for the inequality 3x + 4y > 60 is shown. A graph shows number of math problems on the x-axis, from 0 to 20, and number of science problems on the y-axis, from 0 to 16. A dashed straight line has a negative slope and goes through (4, 12) and (12, 6).Which statement could be true in this situation?Alexia completed 8 math problems and 9 science problems.Alexia completed 4 math problems and 6 science problems.Alexia completed 20 math problems and 10 science problems.Alexia completed no math problems and 15 science problems.

To assess the claims about the number of math and science problems Alexia completed, we start by analyzing the inequality:

\[ 3x + 4y > 60 \]

Where:

  • \(x\) = number of math problems
  • \(y\) = number of science problems

Analyzing the Boundary Line:

The boundary line \(3x + 4y = 60\) has the following points specified:

  • (4, 12): \[ 3(4) + 4(12) = 12 + 48 = 60 \quad \text{(on the boundary line)} \]
  • (12, 6): \[ 3(12) + 4(6) = 36 + 24 = 60 \quad \text{(on the boundary line)} \]

Since the inequality is strict, we are interested in values above this line.

Evaluating Each Statement:

  1. Alexia completed 8 math problems and 9 science problems: \[ 3(8) + 4(9) = 24 + 36 = 60 \quad \text{(not valid because } 60 \not> 60\text{)} \]

  2. Alexia completed 4 math problems and 6 science problems: \[ 3(4) + 4(6) = 12 + 24 = 36 \quad \text{(valid, since } 36 < 60\text{, which means it is below the line)} \]

  3. Alexia completed 20 math problems and 10 science problems: \[ 3(20) + 4(10) = 60 + 40 = 100 \quad \text{(valid, since } 100 > 60\text{)} \]

  4. Alexia completed no math problems and 15 science problems: \[ 3(0) + 4(15) = 0 + 60 = 60 \quad \text{(not valid because } 60 \not> 60\text{)} \]

Conclusion:

  • From our evaluations, the statements that could be valid based on meeting the criteria of \(3x + 4y > 60\) are:
    • Alexia completed 20 math problems and 10 science problems.

The statement "Alexia completed 4 math problems and 6 science problems" is invalid since it is below the line, while all others either sit on the line or do not exceed the minimum time criteria.

Thus, the only statement that could be true regarding the amount of time spent on her problems is:

  • Alexia completed 20 math problems and 10 science problems.