Select the correct answer.
Jackson has a gift card for [tex]$120$[/tex] to use at a department store. He wants to purchase 2 shirts and 3 pairs of pants with his gift card. If [tex]$x$[/tex] represents the cost of the shirts and [tex]$y$[/tex] represents the cost of the pants, what are the domain and range of this relationship?
A. domain: [tex]$0 \leq x \leq 60$[/tex]
range: [tex]$0 \leq y \leq 40$[/tex]
B. domain: [tex]$0 \leq x \leq 40$[/tex]
range: [tex]$0 \leq y \leq 60$[/tex]
C. domain: [tex]$x \geq 0$[/tex]
range: [tex]$y \geq 0$[/tex]
D. domain: [tex]$0 \leq x \leq 120$[/tex]
range: [tex]$0 \leq y \leq 120$[/tex]
Answer (2)
Set up the inequality 2 x + 3 y ≤ 120 to represent the total cost of shirts and pants.
Determine the domain by setting y = 0 and solving for x : 2 x ≤ 120 , which gives x ≤ 60 . Thus, 0 ≤ x ≤ 60 .
Determine the range by setting x = 0 and solving for y : 3 y ≤ 120 , which gives y ≤ 40 . Thus, 0 ≤ y ≤ 40 .
The domain is 0 ≤ x ≤ 60 and the range is 0 ≤ y ≤ 40 , so the answer is A .
Explanation
Setting up the Inequality Let x be the cost of a shirt and y be the cost of a pair of pants. Jackson wants to purchase 2 shirts and 3 pairs of pants with his gift card for $120. This can be represented by the inequality:
The Inequality 2 x + 3 y ≤ 120
Finding the Domain The domain represents the possible values of x (the cost of shirts). Since the cost cannot be negative, xg e q 0 . To find the maximum value of x , we assume that the cost of the pants is 0, so y = 0 . Then the inequality becomes:
Solving for x 2 x + 3 ( 0 ) ≤ 120
2 x ≤ 120
x ≤ 60
Domain So the domain is 0 ≤ x ≤ 60 .
Finding the Range The range represents the possible values of y (the cost of pants). Since the cost cannot be negative, y g e q 0 . To find the maximum value of y , we assume that the cost of the shirts is 0, so x = 0 . Then the inequality becomes:
Solving for y 2 ( 0 ) + 3 y ≤ 120
3 y ≤ 120
y ≤ 40
Range So the range is 0 ≤ y ≤ 40 .
Conclusion Therefore, the domain is 0 ≤ x ≤ 60 and the range is 0 ≤ y ≤ 40 .
Examples
Imagine you're planning a small party and have a budget of $120 for snacks. You want to buy both bags of chips (x) and boxes of cookies (y). If each bag of chips costs $2 and each box of cookies costs $3, the inequality 2 x + 3 y ≤ 120 models your spending. The domain tells you the maximum number of chip bags you can buy if you buy no cookies, and the range tells you the maximum number of cookie boxes you can buy if you buy no chips. Understanding these limits helps you plan your party effectively without overspending.
The domain for the cost of shirts (x) is 0 ≤ x ≤ 60 , while the range for the cost of pants (y) is 0 ≤ y ≤ 40 . Therefore, the correct answer is A: domain: 0 ≤ x ≤ 60 , range: 0 ≤ y ≤ 40 .
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