Let h = number of hours. The total cost of the repair at the dealer and at the local mechanic is calculated the same way. It is the price of the parts plus the cost of labor. The cost of labor is the number of hours multiplied by the labor cost per hour.
Part a)
Dealership:
cost of parts = 24
cost of labor = 99h
total cost = 99h + 24
Local mechanic:
cost of parts = 45
cost of labor = 89h
total cost = 89h + 45
The only variable we have is h, the number of hours. We set the total costs equal and solve for h, the number of hours. That will tell us the number of hours for which the total costs are the same.
99h + 24 = 89h + 45
10h = 21
h = 2.1
Answer for a): 2.1 hours
Part b)
We want to know the number of hours for which the dealership costs less. We use the expressions for costs, but now we set the dealership as costing less than the local dealer. We set up an inequality and solve for h.
99h +24 < 89h + 45
10h < 21
h < 2.1
Answer for b): For less than 2.1 hours of labor, the dealership costs less.
Explanation: The dealership has a lower fixed cost since the parts cost less at the dealership. The labor costs more per hour at the dealership. If the repair took no time at all, the dealership would cost only $24 vs $45 at the local mechanic, so the total bill would be $21 less at the dealership. For every hour of labor at the dealership, you pay $10 more in labor costs, so after 2.1 hours, the two costs are the same, and for more than 2.1 hours, the dealership costs more.
