An office building containing 96,000 square feet of space is to be made into apartments. There will be at most 15 one-bedroom units, each with 800 square feet of space. The remaining units, each with 1200 square feet of space, will have two bedrooms. Rent for each one-bedroom unit will be $650 and for each two-bedroom unit will be $900. Let x represent the number of one-bedroom apartments and y represent the number of two-bedroom apartments. Which system of inequalities represents the number of apartments to be built?

a. x>_15, y>_0,650x +900y<_96,000
b. x<_15, y>_0, 800x + 1200y>_96,000
c. x<_650, y<_900, 800x+1200y<_96,000
d. x<_15, y>_0,800x+1200y<_96,000

I don't understand your symbols, but d looks right.

Thanks you were a great help

The correct answer is d. x<_15, y>_0, 800x+1200y<_96,000.

Let's see why this is the correct answer.

We need to represent the number of one-bedroom apartments as x and the number of two-bedroom apartments as y.

The first condition is that there can be at most 15 one-bedroom units, so x has to be less than or equal to 15. Therefore, the first inequality is x<_15.

The second condition is that there can be any number of two-bedroom units, so y can be any non-negative integer. Therefore, y has to be greater than or equal to 0. This gives us the second inequality, y>_0.

The third condition is that the total square footage of all the apartments cannot exceed 96,000 square feet. The total square footage of the one-bedroom apartments is 800x, and the total square footage of the two-bedroom apartments is 1200y. So the total square footage can be represented as 800x + 1200y. This total square footage has to be less than or equal to 96,000. Therefore, the third inequality is 800x + 1200y<_96,000.

Putting it all together, the system of inequalities representing the number of apartments to be built is:
x<_15, y>_0, 800x+1200y<_96,000.

To determine the correct system of inequalities to represent the number of apartments to be built, we need to consider the given information.

Let's analyze the given information step by step:

1. The office building contains 96,000 square feet of space.
This means the total available space is limited to 96,000 square feet.

2. The number of one-bedroom units should be at most 15, with each unit having 800 square feet of space.
Therefore, the total square footage used by the one-bedroom units can be calculated as 15 * 800 = 12,000x.

3. The remaining units will have two bedrooms, with each unit occupying 1200 square feet of space.
Hence, the total square footage used by the two-bedroom units can be calculated as (Total Space) - (One-bedroom Space) = 96,000 - 12,000x.

Now, let's consider the rental prices:

4. Each one-bedroom unit will have a rent of $650.
This means the revenue from the one-bedroom apartments can be expressed as 650x.

5. Each two-bedroom unit will have a rent of $900.
Hence, the revenue from the two-bedroom apartments can be expressed as 900y.

To summarize, we have the following constraints:

- The total square footage used by both types of apartments should not exceed the available space of 96,000 square feet: 800x + 1200y ≤ 96,000.

- The number of one-bedroom apartments should be at most 15: x ≤ 15.

- The number of two-bedroom apartments can be any non-negative number, as there is no stated limit: y ≥ 0.

Now, let's compare these constraints with the given system of inequalities options:

a. x ≥ 15, y ≥ 0, 650x + 900y ≤ 96,000
This option accurately represents the constraints described above.

b. x ≤ 15, y ≥ 0, 800x + 1200y ≥ 96,000
This option incorrectly includes the opposite inequality (≥) for the revenue constraint.

c. x ≤ 650, y ≤ 900, 800x + 1200y ≤ 96,000
This option sets arbitrary limits on the variables x and y, which are not mentioned in the given information.

d. x ≤ 15, y ≥ 0, 800x + 1200y ≤ 96,000
This option accurately represents the constraints described above.

Therefore, the correct system of inequalities that represents the number of apartments to be built is option a.