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Alina spent no more than [tex]$45 on gas for a road trip. The first gas station charged $[/tex]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, [tex]\( x \)[/tex], the number of gallons of gas she bought at the second station, [tex]\( y \)[/tex], and the total amount she paid? What are the possible values of [tex]\( y \)[/tex]?

A. [tex]\( 3.5x + 4y \geq 45, \quad y \geq 11.25 \)[/tex]

B. [tex]\( 3.5x + 4y \geq 45, \quad 0 \leq y \leq 10.375 \)[/tex]

C. [tex]\( 3.5x + 4y \leq 45, \quad 0 \leq y \leq 11.25 \)[/tex]

D. [tex]\( 3.5x + 4y \leq 45, \quad y \geq 10.375 \)[/tex]

Sagot :

GinnyAnswer
Let's break down the problem step-by-step to find the correct inequality and possible values of [tex]\( y \)[/tex].

1. Identify the given information:
- Maximum amount Alina spent on gas: \[tex]$45. - Cost per gallon at the first gas station: \$[/tex]3.50.
- Cost per gallon at the second gas station: \[tex]$4.00. 2. Formulate the inequality: - Let \( x \) be the number of gallons bought at the first gas station. - Let \( y \) be the number of gallons bought at the second gas station. - The total amount spent on gas cannot exceed \$[/tex]45, so we can write the inequality as:
[tex]\[ 3.5x + 4y \leq 45 \][/tex]

3. Determine the possible values of [tex]\( y \)[/tex]:
- First, recognize that [tex]\( x \)[/tex] and [tex]\( y \)[/tex] must be non-negative since you can't buy a negative amount of gas:
[tex]\[ x \geq 0 \quad \text{and} \quad y \geq 0 \][/tex]

- To find the maximum value of [tex]\( y \)[/tex], consider the scenario where [tex]\( x = 0 \)[/tex]:
[tex]\[ 3.5(0) + 4y \leq 45 \implies 4y \leq 45 \implies y \leq \frac{45}{4} \implies y \leq 11.25 \][/tex]

- To find the minimum value of [tex]\( y \)[/tex], recognize again that y must be non-negative:
[tex]\[ y \geq 0 \][/tex]

Therefore, the inequality and the range of possible values of [tex]\( y \)[/tex] are:

[tex]\[ 3.5x + 4y \leq 45, \quad 0 \leq y \leq 11.25 \][/tex]

Hence, the correct answer is:
[tex]\[ \boxed{3.5 x+4 y \leq 45,0 \leq y \leq 11.25} \][/tex]
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