The Correct Answer and Explanation is:
The correct answer is C. 3.5x + 4y ≤ 45, 0 ≤ y ≤ 11.25.
Here is a detailed explanation of how to arrive at this answer.
First, we need to translate the word problem into a mathematical inequality. The problem states that Alina spent “no more than $45” on gas. The phrase “no more than” means the total amount spent must be less than or equal to $45. This gives us the inequality symbol ≤.
Next, let’s define the variables and set up an expression for the total cost.
- x represents the number of gallons of gas bought at the first station. The cost at this station is $3.50 per gallon, so the total amount spent here is 3.50x.
- y represents the number of gallons of gas bought at the second station. The cost at this station is $4.00 per gallon, so the total amount spent here is 4.00y or simply 4y.
The total amount Alina spent on gas is the sum of the costs from both stations: 3.5x + 4y.
Combining this expression with the spending limit, we get the complete inequality that relates the variables to the total amount paid:
3.5x + 4y ≤ 45
This first part of the solution eliminates options A and B, which incorrectly use the ≥ symbol.
The second part of the question asks for the possible values of y. To find the range for y, we must consider its minimum and maximum possible values based on the inequality and real world constraints.
The minimum value for y is 0. Since y represents the number of gallons of gas, Alina cannot buy a negative amount. The least she could buy from the second station is zero gallons. Therefore, y ≥ 0.
To find the maximum possible value for y, we must consider the scenario where Alina spends as much of her budget as possible on gas from the second station. This happens when she buys the least possible amount of gas from the first station. The minimum amount of gas she can buy from the first station is x = 0 gallons.
We can substitute x = 0 into our inequality to solve for the maximum value of y:
3.5(0) + 4y ≤ 45
0 + 4y ≤ 45
4y ≤ 45
Now, divide both sides by 4 to isolate y:
y ≤ 45 / 4
y ≤ 11.25
This tells us that the maximum number of gallons Alina can buy from the second station is 11.25.
By combining the minimum (y ≥ 0) and maximum (y ≤ 11.25) values, we find the complete range of possible values for y:
0 ≤ y ≤ 11.25
Therefore, the correct choice must have the inequality 3.5x + 4y ≤ 45 and the range 0 ≤ y ≤ 11.25. This perfectly matches option C.
