Question

The toll to a bridge is $3.50. A three-month pass costs $18.00 and reduces the toll to $0.50. A six-month pass costs $55 and permits crossing the bridge for no additional fee. How many crossings per three-month period does it take for the three-month pass to be the best deal (assuming a consistent number of crossings for all three-month periods)? Let x represent the number of crossings in a three month period for what values of x is the three month pass the best deal? Make the answer an inequality or a compound inequality

Answer 1

Answer: 0 ≤ x < 74

This is the same as x < 74 when x is nonnegative.

In either case, x is an integer.

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Explanation.

x = number of crossings in a 3 month period

The three-month pass charges a start fee of $18. Then tacks on $0.50 per crossing. Doing x number of these crossings means we add 0.50x dollars onto the $18 initially.

The total cost of the three-month pass is 18+0.50x dollars.

The six-month pass costs $55 with no extra fees. You can cross as many times as you want, and only pay $55 total for the entire three month period.

To find out when the three-month pass is better than the six-month pass, we will form this inequality: 18+0.5x < 55. The stuff on the left is the three-month pass, and the right hand side is the six-month pass.

Let's solve for x.

18+0.5x < 55

0.5x < 55-18

0.5x < 37

x < 37/0.5

x < 74

If you make fewer than 74 crossings, then the three-month pass is the cheaper option.

For example, if you do x = 60 crossings then,

18+0.5x = 18+0.5*60 = 18+30 = 48 dollars is the total, which beats the $55 from the other plan.

On the other hand, if you do x = 90 crossings, then,

18+0.5x = 18+0.5*90 = 18+45= 63 dollars is the total, which is now more expensive compared to the $55.

Because x is a nonnegative number, we can expand x < 74 into 0 ≤ x < 74 to be more clear what values x can take on.

Both x < 74 and 0 ≤ x < 74 are valid answers. If you go with the first one, then be sure to state that x cannot be negative. In either case, x is an integer. We cannot have something like x = 2.189 crossings.