Question 1182928
price per gallon of gas with 7 dollar car wash is 3.19.
price per gallon of gas without the car wash is 3.39.


y = total price
x = number of gallons of gas.
price of car wash is 7.00.

total cost with car wash is y = 7 + 3.19 * x
total cost without car wash is y = 3.39 * x


it is worth it to buy the car wash when the total price with the car wash is less than the total price without the car wash.


the inequality for that is:


7 + 3.19 * x < 3.39 * x


subtract 3.19 * x from both sides of that equation to get:
7 <= 3.39 * x - 3.19 * x
simplify to get:
7 < .20 * x
divide both sides by .2 to get:
7/.20 < x
simplify to get:
35 < x
this is the same as:
x > 35


if the number of gallons of gas is greater than 35, then it is cheaper to buy the gas with the car wash than without.


if you buy exactly 35 gallons of gas, you will pay the same for with the car wash and without the car wash.
7.00 + 3.19 * 35 = 118.65
3.39 * x = 118.65.


if you buy 30 gallons of gas, you will pay less for without the car wash than with the car wash.
7.00 + 3.19 * 30 = 102.7
3.39 * 30 = 101.7


if you buy 40 gallons of gas, you will pay less for with the car wash than without the car wash.
7.00 + 3.19 * 40 = 134.6
3.39 * 40 = 135.6


if the car wash costs 2 dollars instead of 7, then the equations are:


y = 2 + 3.19 * x for with the car wash.
y = 3.39 * x for without the car wash.


the total cost for with the car wash will be less than the total cost without the car wash when 2.00 + 3.19 * x <= 3.39 * x


subtract 3.19 * x from both sides to get:
2 < .20 * x
divide both sides by .2 to get:
2/.2 < x
simplify to get:
10 < x
this is the same as x > 10


the break even point is when x = 10
you pay less without the car wash when x < 10
you pay less with the car wash when x > 10


when x = 10, you pay the same with or without the car wash.
2 + 3.19 * 10 = 33.9
3.39 * 10 = 33.9


when x = 9, you pay less without the car wash.
2 + 3.19 * 9 = 30.71
3.39 * 9 = 30.51


when x = 11, you pay less with the car wash.
2 + 3.19 * 11 = 37.09
3.39 * 11 = 37.29