Answer
In order for the mechanic to give the customer the estimate, he must be charging for working $at$ $least$ $1.5$ hours, up to a $maximum$ $of$ $3.5$ hours.
Work Step by Step
To solve this exercise, we must first model the total cost the mechanic is charging the customer: $$Cost_{total} = 175 + 34h$$ where $h$ represents the total amount of hours that the mechanic works on the car. Since the estimate that the mechanic gives the customer is at least 226 dollars and at most 294, we can model this in the following manner: $$226 \leq Cost_{total} \leq 294$$ $$226 \leq 175 + 34h \leq 294$$ By solving for $h$: $$226 - 175 \leq 34h \leq 294 - 175$$ $$51 \leq 34h \leq 119$$ $$1.5 \leq h \leq 3.5$$ which means that, in order for the mechanic to give the customer the estimate, he must be charging for working $at$ $least$ $1.5$ hours, up to a $maximum$ $of$ $3.5$ hours.
