Question

Brennan recently spent $160 on woodworking tools so he can make and sell personalized mailboxes. Each mailbox costs $14 to make, and Brennan sells the mailboxes for $40 each.
Brennan wants to find how many mailboxes, x, he must make and sell in order to make a profit. Which inequality can he use?

Answer 1

To determine how many mailboxes Brennan needs to make and sell to generate a profit, we first need to establish his costs and revenues.

1. **Initial Investment:** Brennan spent $160 on woodworking tools (fixed costs).
2. **Cost per Mailbox:** Each mailbox costs $14 to make (variable cost).
3. **Selling Price per Mailbox:** Each mailbox is sold for $40.

**Total Cost (TC) when making x mailboxes:**
\( TC = 160 + 14x \)

**Total Revenue (TR) from selling x mailboxes:**
\( TR = 40x \)

**To achieve a profit, the total revenue must be greater than the total cost:**
\( TR > TC \)

Putting this into an inequality:
\[ 40x > 160 + 14x \]

To find the number of mailboxes \( x \) he needs to sell to make a profit, we can rearrange the inequality:
1. Subtract \( 14x \) from both sides:
\[ 40x - 14x > 160 \]
\[ 26x > 160 \]

2. Divide both sides by 26:
\[ x > \frac{160}{26} \]
\[ x > \frac{80}{13} \]
\[ x > 6.15 \]

Since Brennan can't sell a fraction of a mailbox, he must sell at least 7 mailboxes.

Thus, the inequality Brennan can use to find how many mailboxes he must make and sell to earn a profit is:

**Answer:**
\( 40x > 160 + 14x \)

This simplifies to:
\( 26x > 160 \), or \( x > \frac{160}{26} \).