Problem 3

Question

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

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 2

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 3

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 4

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

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.