Answer
In order to make the 3-mo pass a better deal, the customer has to make $less$ $than$ $7.5$ passes in a 3-mo period. Since the passes are counted in whole numbers, we can also say that the customer can only pass through the toll $7$ $times$ $or$ $less$ if he or she is to use the 3-mo package as a better deal than the 6-mo package.
Work Step by Step
To solve this exercise, we must first model the cost of each toll package: $$Pass_{(3mo)} = 7.5 + 0.5T$$ $$Pass_{(6mo)} = 30$$ where $T$ represents the amount of times a customer passes the toll. Since the exercise asks for the amount of tolls it takes to make the 3-mo pass a better deal, we can model this in the following manner: $$Pass_{(3mo)} \lt Pass_{(6mo)}$$ $$7.5 + 0.5T \lt 30$$ However, we must be aware that we're comparing the 3-mo system to a 6-mo system. Therefore, we're actually comparing twice the cost of the 3-mo pass to the 6mo pass: $$2\times(7.5 + 0.5T) \lt 30$$ $$15 + T \lt 30$$By solving for $T$: $$T \lt 30 - 15$$ $$T \lt 15$$ This means that, in order to make the 3-mo pass a better deal, the customer has to make less than 15 passes through the toll in a 6-month period., which is to say, $less$ $than$ $7.5$ passes in a 3-mo period. Since the passes are counted in whole numbers, we can also say that the customer should pass through the toll $7$ $times$ $or$ $less$ if he or she is to use the 3-mo package as a better deal than the 6-mo package.
