A businesswoman wants to determine the difference between the costs of owning and leasing an automobile. She can lease a car for ​$420 a per month​ (on an annual​ basis). Under this​ plan, the cost per mile is ​$0.08. If she were to purchase the​ car, the fixed annual expense would be ​$4600​, and other costs would amount to ​$0.10 per mile. What is the least number of miles she would have to drive per year to make leasing no more expensive than​ purchasing?

First we need to find the total annual costs for both the plans.

In case of leasing: 
Fixed monthly cost = $420
So, yearly cost = 420 x 12 = $5040
Cost per mile = $0.08

For x miles driven, the cost per year for leasing will be = 5040 + 0.08x

In case of purchasing:
Fixed yearly cost = $4600
Cost per mile = 0.10

For x miles driven, the cost per year for leasing will be = 4600 + 0.10x

We want to find for what number of miles will the cost of leasing will be no more expensive than the cost of purchasing.

So,

Cost of leasing ≤ Cost of purchasing
5040 + 0.08x ≤ 4600 + 0.10x
440 ≤ 0.02x
22000 ≤ x

Thus, if if the number of miles driven are equal to or less than 22,000 leasing will be no more expensive than purchasing.