Final answer:
The inequality that represents the number of hours, x, Natasha can bike for the cost of purchasing a bike to be less than renting is x > 5.29. She should purchase the bike if she plans to bike for more than approximately 5.29 hours.
Step-by-step explanation:
The question asks for an inequality representing the number of hours, x, for which the cost of purchasing a mountain bike is less than the cost of renting a bike and helmet together for x hours. To determine this, we need to compare the cost of purchase plus the hourly rental of the helmet to the cost of renting both items together.
Let's denote the number of hours Natasha plans to go mountain biking as x. If Natasha purchases the bike and rents the helmet, her cost would be the constant purchase price of the bike, $185, plus $6 per hour for the helmet. So the cost in this case is represented by the equation 185 + 6x. On the other hand, if she rents both the bike and the helmet, her cost is represented by the equation 41x.
To find the inequality that represents when purchasing the bike is cheaper than renting, we set up the inequality 185 + 6x < 41x. Simplifying this, we get:
185 < 41x - 6x
185 < 35x
x > 185/35
x > 5.2857
So Natasha should purchase the bike if she plans to bike for more than approximately 5.29 hours.