Question 1190430
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The "unlimited" miles means that the company doesn't charge per mile in this scenario. All they get is the $60 per day.
If you only rent for one day, then all they get is the $60. This applies to plan B only of course.


x = number of miles
y = total cost


If you only plan to do a 1 day rental, then,
Plan A has the equation y = 0.19x+40
The 0.19x is the extra 19 cents (aka $0.19) per mile on top of the $40 initially paid


Plan B has the equation y = 60 for 1 day rentals.
The x is not involved because again they don't charge per mile. The cost is simply 60 dollars.


Both cost equations involve y
Let's apply substitution to find x
y = 60
0.19x+40 = 60 ... y replaced with 0.19x+40
0.19x = 60-40
0.19x = 20
x = 20/0.19
x = 105.263 approximately
This rounds to 105.3 when rounding to one decimal place.
The two costs are roughly the same when driving 105.3 miles for 1 day rentals.


If you traveled 105 miles in one day, then plan A costs you
y = 0.19x+40
y = 0.19*105+40
y = 19.95+40
y = 59.95
which is just barely under the $60 mark. This shows that plan A is the better choice for this mileage.


If you traveled 106 miles in one day, then plan A costs you
y = 0.19x+40
y = 0.19*106+40
y = 20.14+40
y = 60.14
You are paying 14 cents more than if you went with plan B.
For any larger x values, plan A will get more expensive and it's better to go for plan B.
Intuitively this is what we'd expect because plan A's cost grows based on the number of miles driven, while B has a fixed cost.


Those are two examples to show that the boundary is 105.3
I'll let you play with other x values. I recommend making a table to compare costs.


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Answers:


For what range of miles will Plan B save you money for a 1 day rental? <font color=red>Anything larger than 105.3 miles</font> 
For what range of miles will Plan A save you money for a 1 day rental? <font color=red>Anything between 0 and 105.3 miles</font>


In terms of inequalities, the first statement in red is the same as saying {{{x > 105.3}}} while the second statement would translate to {{{0 <= x <= 105.3}}}, where x is the number of miles driven. Also, this all applies to one day rentals only.


If you require more accuracy, then change the 105.3 to something like 105.26 or perhaps 105.263 or whatever you prefer to round to; there are infinitely many ways to write this value because we're dealing with a rounded approximation. 
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