A very old vending machine accepts only nickels (n) and dimes (d). Candy costs up to $0.50, but sometimes the machine will dispense candy without any coins being inserted into the machine. Which inequality shows all of the ways to obtain a candy bar from the machine?
To determine the inequality that shows all of the ways to obtain a candy bar from the vending machine, we need to consider the possible combinations of nickels and dimes that would be sufficient to cover the cost of candy.
The cost of candy can be anywhere up to $0.50, which means it could be $0.50 itself or any amount less than $0.50. We can represent this using the variable 'c' for the cost of candy.
Now, let's consider the different combinations of nickels (n) and dimes (d) that can be used to cover the cost.
1. If the cost of candy is exactly $0.50, we can use two dimes to pay for it. This can be represented by the inequality: 2d ≤ c.
2. If the cost of candy is less than $0.50, we can use a combination of nickels and dimes. Let's assume we need 'x' nickels and 'y' dimes to cover the cost. The value of 'x' could range from 0 to (0.50-c)/0.05, which represents the maximum possible number of nickels we can use. The value of 'y' could range from 0 to (0.50-c)/0.10, which represents the maximum possible number of dimes we can use. Hence, we have the inequalities:
0 ≤ x ≤ (0.50 - c)/0.05 and
0 ≤ y ≤ (0.50 - c)/0.10.
To summarize, the inequality that shows all of the ways to obtain a candy bar from the vending machine is:
2d ≤ c or 0 ≤ x ≤ (0.50 - c)/0.05 and 0 ≤ y ≤ (0.50 - c)/0.10, where 'c' represents the cost of candy.