At a campground, kayak rental company A charges a fixed amount of $10 plus $8 per hour for renting a one-person kayak. Company B charges a fixed amount of $6 plus $10 per hour for a one-person kayak. Write and solve an inequality representing the number of hours where Option A will be the cheaper rental company. Let x represent the number of hours.
For company A, the total cost for renting a kayak for x hours is given by:
Cost for Company A = $10 + $8x
For company B, the total cost for renting a kayak for x hours is given by:
Cost for Company B = $6 + $10x
If Option A is the cheaper rental company, then the cost for Company A must be less than the cost for Company B. Therefore, we can write the following inequality:
$10 + $8x < $6 + $10x
To solve for x, we can start by subtracting $6 from both sides of the inequality:
$4 + $8x < $10x
Then, subtract $8x from both sides of the inequality:
$4 < $2x
Finally, divide both sides of the inequality by $2:
2 < x
So the inequality representing the number of hours where Option A will be the cheaper rental company is:
x > 2
To represent the number of hours where Option A will be the cheaper rental company, we need to compare the total cost for a one-person kayak from both companies.
For Option A, the total cost in dollars can be calculated using the formula:
Cost for Option A = Fixed amount + (Hourly rate × Number of hours)
Cost for Option A = $10 + ($8 × x)
For Option B, the total cost in dollars can be calculated using the formula:
Cost for Option B = Fixed amount + (Hourly rate × Number of hours)
Cost for Option B = $6 + ($10 × x)
To find the number of hours where Option A will be cheaper, we need to set up an inequality by comparing the total cost for both options. We want the situation where Option A is cheaper, so we can write:
Cost for Option A < Cost for Option B
$10 + ($8 × x) < $6 + ($10 × x)
Now, we can solve this inequality to find the range of hours where Option A is cheaper:
$10 + ($8 × x) < $6 + ($10 × x)
10 + 8x < 6 + 10x
Now, let's isolate the x term on one side:
8x - 10x < 6 - 10
-2x < -4
Finally, divide both sides by -2 (remembering to flip the inequality sign when dividing by a negative number):
x > -4 / -2
x > 2
Therefore, the range of hours where Option A will be the cheaper rental company is when x is greater than 2.
To find out when Option A (company A) will be the cheaper rental company, we need to compare the total cost of renting a one-person kayak from both companies.
For company A, the total cost can be represented by the equation:
Total cost = $10 + $8/hour
For company B, the total cost can be represented by the equation:
Total cost = $6 + $10/hour
To determine when company A is cheaper, we need to find the values of x (the number of hours) for which the total cost of renting from company A is less than the total cost of renting from company B.
So, the inequality representing this scenario is:
10 + 8x < 6 + 10x
Now, let's solve the inequality step-by-step:
10 + 8x < 6 + 10x
(Subtract 6 from both sides)
4 + 8x < 10x
(Subtract 8x from both sides)
4 < 2x
(Divide both sides by 2)
2 < x
Therefore, for Option A to be the cheaper rental company, the number of hours (x) must be greater than 2.