To solve this problem, let's carefully analyze the components of the plumber's compensation and create an appropriate inequality to represent the number of hours she needs to work to make at least [tex]$\$[/tex]50,000[tex]$.
### Step-by-Step Solution:
1. Understand the Components of Compensation:
- The plumber earns $[/tex]\[tex]$50$[/tex] per hour of work.
- She also receives a [tex]$\$[/tex]5,000[tex]$ profit-share at the end of the year.
2. Express Total Annual Compensation:
- Let $[/tex]n[tex]$ be the number of hours the plumber works in a year.
- her compensation per hour is $[/tex]\[tex]$50$[/tex], so for [tex]$n$[/tex] hours, she earns [tex]$50n$[/tex] dollars.
- Adding the profit-share of [tex]$\$[/tex]5,000[tex]$, her total compensation will be $[/tex]50n + 5,000[tex]$ dollars.
3. Setting Up the Inequality:
- We need to determine the number of hours, $[/tex]n[tex]$, required to make at least $[/tex]\[tex]$50,000$[/tex] in total compensation.
- This is represented by the inequality [tex]$50n + 5,000 \geq 50,000$[/tex].
4. Solve the Inequality:
- Subtract [tex]$\$[/tex]5,000[tex]$ from both sides of the inequality to isolate the term involving $[/tex]n[tex]$:
\[
50n + 5,000 \geq 50,000 \\
50n \geq 50,000 - 5,000 \\
50n \geq 45,000
\]
- Divide both sides by $[/tex]50[tex]$ to solve for $[/tex]n[tex]$:
\[
n \geq \frac{45,000}{50} \\
n \geq 900
\]
### Final Answer:
The inequality that represents the number of hours, $[/tex]n[tex]$, she needs to work to earn at least $[/tex]\[tex]$50,000$[/tex] is:
[tex]\[ \boxed{50,000 \leq 50n + 5,000} \][/tex]
### Conclusion:
The correct answer is (A) [tex]$50,000 \leq 50n + 5,000$[/tex] which means the plumber needs to work at least 900 hours to achieve at least [tex]$\$[/tex]50,000$ total compensation for the year.
