Question

Problem 3
Demetrius can spend as much as
​$50 on shirts.



Shirts,
s, cost
$16 each at a nearby store.



Which inequality represents this situation?

Answer 1

Choose the inequality that matches this situation.



Company A charges $312 to power-wash the siding of a house plus $12 for each window. Company B charges $36 per window, and the price includs the siding. How many windows must a house have to make the total cost from Company A be less expensive than Company B?

Answer 2

make an inequality that shows company a being less than company b and explain why its less

Answer 3

To represent the situation where Demetrius can spend as much as $50 on shirts that cost $16 each, we can set up the inequality as follows:

Let \( s \) represent the number of shirts Demetrius can buy.

The total cost of the shirts can be represented as \( 16s \) (since each shirt costs $16).

The inequality that represents the situation is:

\[
16s \leq 50
\]

This means that the total cost of the shirts purchased must be less than or equal to $50.

Answer 4

To determine how many windows must a house have for Company A's total cost to be less expensive than Company B's total cost, we can set up inequalities based on the given information.

**For Company A:**

- Fixed cost for power-washing the siding: $312
- Additional cost for each window: $12 per window

So, the total cost for Company A can be represented as:
\[
C_A = 312 + 12w
\]
where \( w \) is the number of windows.

**For Company B:**

- Cost per window (which includes siding): $36

So, the total cost for Company B can be represented as:
\[
C_B = 36w
\]

**We want to find the inequality where Company A's cost is less than Company B's cost:**
\[
312 + 12w < 36w
\]

Now, we can simplify this inequality:

1. Subtract \( 12w \) from both sides:
\[
312 < 36w - 12w
\]
2. Combine like terms:
\[
312 < 24w
\]
3. Divide both sides by 24:
\[
\frac{312}{24} < w
\]
4. Simplify:
\[
13 < w
\]

This can be written as:
\[
w > 13
\]

Thus, a house must have more than 13 windows for Company A to be less expensive than Company B.

Answer 5

To illustrate the situation where Company A is less expensive than Company B, we can set up an inequality as described previously. Here's how we can express it:

### Setup of Inequalities

1. **Company A** charges $312 to power-wash the siding of a house plus $12 for each window:
\[
C_A = 312 + 12w
\]
Where \( C_A \) is the total cost from Company A and \( w \) is the number of windows.

2. **Company B** charges $36 per window, which includes the price of siding:
\[
C_B = 36w
\]
Where \( C_B \) is the total cost from Company B.

### Inequality

To find when Company A's total cost is less than Company B's total cost, we set up the following inequality:
\[
312 + 12w < 36w
\]

### Explanation of the Inequality

1. **Breaking it down:**
- **Left-hand side (\(312 + 12w\)):** This represents the total cost of Company A, which consists of a fixed fee of $312 for power-washing and an additional charge of $12 for each window.
- **Right-hand side (\(36w\)):** This represents the total cost from Company B, which charges $36 per window, including the cost of power-washing.

2. **Why it's less:**
- The goal of this inequality is to identify how many windows \( w \) can be added to Company A's cost while remaining cheaper than Company B's cost.
- To determine when Company A's total cost is less, we're essentially finding the threshold (or breakpoint) where \( C_A < C_B \).

### Solution

To solve the inequality:
1. Subtract \( 12w \) from both sides:
\[
312 < 36w - 12w
\]
Simplifies to:
\[
312 < 24w
\]

2. Divide both sides by 24:
\[
\frac{312}{24} < w
\]
Simplifying gives:
\[
13 < w
\]

Thus, we conclude:
\[
w > 13
\]

### Interpretation

This means that if there are more than **13 windows**, the total cost charged by **Company A** will be less than that of **Company B**. Therefore, for **a house with at least 14 windows**, choosing **Company A** will yield a lower total expense.