A patio furniture company makes a new porch umbrella. The company earns [tex]\$75[/tex] for each umbrella sold. It costs [tex]\$16,000[/tex] to design and produce the umbrella. Write an inequality to find how many umbrellas the company must sell to make more than [tex]\$20,000[/tex].

A. [tex]75x - 16000 \ \textgreater \ 20000[/tex]
B. [tex]16000 - 75x \ \textgreater \ 20000[/tex]
C. [tex]75x - 16000 \ \textless \ 20000[/tex]
D. [tex]16000x - 75 \ \textgreater \ 20000[/tex]

To find how many umbrellas the company must sell to make more than [tex]$20,000, we can set up an inequality based on their profit.

1. Define the terms:
- Let $ x $ represent the number of umbrellas sold.
- The company makes $[/tex]75 per umbrella, so the revenue from selling [tex]$ x $[/tex] umbrellas is [tex]$ 75x $[/tex].
- The cost to design and produce the umbrellas is [tex]$16,000.

2. Set up the inequality for the profit:
- Profit is calculated as the revenue minus the cost, which needs to be greater than $[/tex]20,000.
- The inequality is: [tex]$ 75x - 16000 > 20000 $[/tex].

3. Solve the inequality:

Start with the inequality:
[tex]\[
75x - 16000 > 20000
\][/tex]

Add [tex]$16,000 to both sides to isolate the term with $ x $:
\[
75x > 20000 + 16000
\]

Simplify the right side:
\[
75x > 36000
\]

Finally, divide both sides by 75 to solve for $ x $:
\[
x > \frac{36000}{75}
\]

Calculate the division:
\[
x > 480
\]

Therefore, the company must sell more than 480 umbrellas to achieve a profit greater than $[/tex]20,000.

The correct inequality is represented by option (A): [tex]$ 75x - 16000 > 20000 $[/tex].