Algebra - Inequality word problem (hard)?

[Divine]

The toll to a bridge is $3.00. A three-month pass costs $7.50 and reduces the toll to $0.50. A six-month pass costs $30 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?

The answer is: more than 3 and less than 15 crossings per 3 month period.

I don't understand how to produce that answer. I set up a compound inequality to solve:

7.5 + .5x < 3x and 7.5 + .5x < 30

x > 3 and x < 45.

This looks like you would need more than 3 and less than 45 crossings to make the three month pass the better deal. Where is my confusion?

[maggiemagnet]

The toll to a bridge is $3.00. A three-month pass costs $7.50 and reduces the toll to $0.50. A six-month pass costs $30 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?

You need the 3-month pass to be less than each of the other two options for the same period of time. So you need to compare the base rate for six months to the six-month pass for six months and to the three-month pass for six months. So you need two of the three-month passes!

($7.50 /3-mo period)*(2 3-mo periods) + ($0.50 / crossing)*(x crossings) < ($3.00 / crossing)*(x crossings)

($7.50 /3-mo period)*(2 3-mo periods) + ($0.50 / crossing)*(x crossings) < ($30.00 flat fee)

This gives you 15 + 0.5x < 3x and 15 + 0.5x < 30. See how that works out! (Remember to work back to the "crossings per three-month period" part for the actual answer.)