Use an inequality to solve the problem. Be sure to show the inequality and all of your work. A car rental company has two rental rates. Rate 1 is $63 per day plus $0.18 per mile. Rate 2 is $126 per day plus $0.09 per mile. If you plan to rent for one week, how many miles would you need to drive to pay less by taking Rate 2?
let the number of miles driven be x
when is .09x + 126 ≤ .18x + 63 ?
times 100
9x + 12600 ≤ 18x + 6300
-9x ≤ - 6300
x ≥ 700
x >700
To solve this problem using an inequality, we need to compare the total cost of both rates and find the condition where Rate 2 is cheaper. Let's represent the number of miles driven as "m."
The total cost for Rate 1 can be calculated by multiplying the daily rate by the number of days and adding the cost per mile multiplied by the number of miles driven. Since the rental is for one week (7 days), the cost for Rate 1 is:
Cost for Rate 1 = (63 * 7) + (0.18 * m)
The total cost for Rate 2 is calculated in the same way:
Cost for Rate 2 = (126 * 7) + (0.09 * m)
To determine when Rate 2 is cheaper than Rate 1, we can set up the following inequality:
Cost for Rate 2 < Cost for Rate 1
Now let's substitute the expressions for the costs into the inequality:
(126 * 7) + (0.09 * m) < (63 * 7) + (0.18 * m)
Now we can simplify the inequality:
882 + 0.09m < 441 + 0.18m
To solve for "m," we need to isolate the variable on one side of the inequality. First, let's subtract 441 from both sides of the inequality:
882 - 441 + 0.09m < 0.18m
This simplifies to:
441 + 0.09m < 0.18m
Next, let's subtract 0.09m from both sides to eliminate the terms containing "m":
441 < 0.09m
Finally, divide both sides of the inequality by 0.09 to solve for "m":
441 / 0.09 < m
This simplifies to:
4900 < m
Therefore, to pay less by taking Rate 2, you would need to drive more than 4900 miles.