A bounce house rental company A charges a fixed amount of $30 plus $5 per hour for renting a bounce house. Company B charges a fixed amount of $10 plus $15 per hour for renting a bounce house. 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.
To represent the number of hours where Option A will be the cheaper rental company, we need to compare the cost of renting from Company A with the cost of renting from Company B and set up an inequality.
The cost of renting from Company A can be represented as 30 + 5x, where x is the number of hours.
The cost of renting from Company B can be represented as 10 + 15x, where x is the number of hours.
We want to find the number of hours where the cost of renting from Company A is less than the cost of renting from Company B. So, we want to find the values of x that satisfy the inequality:
30 + 5x < 10 + 15x
Now, we can solve this inequality:
30 + 5x - 5x < 10 + 15x - 5x
30 < 10 + 10x
20 < 10x
2 < x
Therefore, the inequality representing the number of hours where Option A will be the cheaper rental company is x > 2.
Patricia is deciding if she wants to join the flower club. If she joins the club she will pay a membership fee of $65. Members will be given a discounted rate for a dozen roses at $10 per dozen. Non-members will pay $18 per dozen roses. How many times would Patricia need to order a dozen roses for the membership to be cheaper than a nonmembership? Set up an inequality for this problem and solve. Is it cheaper to be a member? Why or why not?
Let's assume Patricia orders x dozens of roses.
As a member, Patricia will pay a discounted rate of $10 per dozen. So her cost will be 10x dollars.
As a non-member, Patricia will pay the regular rate of $18 per dozen. So her cost will be 18x dollars.
We want to find the number of times Patricia needs to order a dozen roses for the membership to be cheaper than a non-membership. So we want to find the values of x that satisfy the inequality:
10x < 18x
Now, we can solve this inequality:
18x - 10x > 0
8x > 0
x > 0
Therefore, Patricia needs to order more than 0 dozen roses for the membership to be cheaper than a non-membership.
To determine if it is cheaper to be a member, we need to consider the total cost.
As a member, Patricia has to pay a membership fee of $65 in addition to the cost of the roses.
As a non-member, Patricia doesn't have to pay a membership fee but would pay the regular rate for the roses.
So, for Patricia, it is cheaper to be a member if she orders more than 0 dozen roses. The membership fee is a fixed cost that is offset by the discounted rate for the roses.
explain
Sure! Let's break it down step by step.
First, we need to determine how much Patricia would pay if she is a member. The membership fee is a one-time cost of $65. In addition to that, Patricia will receive the discounted rate of $10 per dozen roses. Let's say she orders x dozens of roses. So, her cost as a member would be 10x dollars.
Next, let's find out how much Patricia would pay if she is not a member. In this case, she doesn't have to pay the membership fee but she will have to pay the regular rate of $18 per dozen roses. So, her cost as a non-member would be 18x dollars.
Now, we want to determine when it is cheaper for Patricia to be a member. We need to find the values of x for which the cost as a member (10x) is less than the cost as a non-member (18x).
To find this, we set up the inequality:
10x < 18x
And solve it:
18x - 10x > 0
8x > 0
Since the coefficient of x is positive, we can divide both sides of the inequality by 8 without changing the direction of the inequality:
x > 0
This means that Patricia needs to order more than 0 dozen roses for the membership to be cheaper. If she orders 0 dozen roses, the cost as a member and as a non-member would be the same.
To determine if it is cheaper to be a member, we need to consider the total cost. As a member, Patricia has to pay a membership fee of $65 in addition to the cost of the roses. As a non-member, she doesn't have to pay the membership fee, but she pays the regular rate for the roses.
So, for Patricia, it is cheaper to be a member if she orders more than 0 dozen roses. The membership fee is a fixed cost that is offset by the discounted rate for the roses.
how to solve
To solve the inequality 10x < 18x, we want to isolate the variable x.
First, subtract 10x from both sides of the inequality:
10x - 10x < 18x - 10x
This simplifies to:
0 < 8x
Next, divide both sides of the inequality by 8:
0/8 < 8x/8
This simplifies to:
0 < x
So the solution to the inequality is x > 0.
This means that Patricia needs to order more than 0 dozen roses for the membership to be cheaper. If she orders 0 dozen roses, the cost as a member and as a non-member would be the same.
Therefore, it is cheaper for Patricia to be a member and pay the membership fee of $65, along with the discounted rate of $10 per dozen roses, rather than paying the regular rate of $18 per dozen roses as a non-member.
Office Max is offering $40 mail in rebate on all Office chairs. George is looking at a few different office chairs that range in price from $195 to $300. How much can he plan to pay after the rebate? Form a compound inequality to represent this problem and solve. Can he stay within his $150 budget for the office chair? Why or why not?
To represent this problem, we can set up a compound inequality that represents the range of prices for the office chairs before the mail-in rebate:
195 ≤ x ≤ 300
Where x represents the price of the office chair.
Now, to find out how much George can plan to pay after the rebate, we subtract the $40 mail-in rebate from each side of the compound inequality:
195 - 40 ≤ x - 40 ≤ 300 - 40
155 ≤ x ≤ 260
This means that after the $40 mail-in rebate, George can plan to pay any amount between $155 and $260 for the office chair.
To determine if George can stay within his $150 budget for the office chair, we compare the price range after the rebate with his budget.
Since his budget is $150, we need to check if $150 is within the range of prices after the rebate:
155 ≤ 150 ≤ 260
This inequality is not satisfied since 150 is less than 155.
Therefore, George cannot stay within his $150 budget for the office chair. The minimum price after the rebate is $155, which is greater than his budget.