On January 1, 2000, there were 175,000 tons of trash in a landfill that had a capacity of 325,000 tons. Each year since then, the amount of trash in the landfill increased by 7,500 tons. If y represents the time, in years, after January 1, 2000, which of the following inequalities describes the set of years where the landfill is at or above capacity?

To determine the set of years when the landfill is at or above capacity, we need to establish an inequality that describes this condition. Let's break down the problem step by step.

1. **Given Information:**
   - Initial amount of trash in the landfill on January 1, 2000: \( 175,000 \) tons.
   - Capacity of the landfill: \( 325,000 \) tons.
   - Annual increase in trash: \( 7,500 \) tons.
   - \( y \) represents the time, in years, after January 1, 2000.

2. **Formulate the Expression for Total Trash:**
   The total amount of trash \( T(y) \) in the landfill \( y \) years after January 1, 2000 can be expressed as:
   \[
   T(y) = 175,000 + 7,500y
   \]

3. **Set Up the Inequality:**
   We need to find the values of \( y \) for which the landfill is at or above its capacity of \( 325,000 \) tons. Therefore, we set up the following inequality:
   \[
   175,000 + 7,500y \geq 325,000
   \]

4. **Solve the Inequality:**
   Subtract \( 175,000 \) from both sides to isolate the term with \( y \):
   \[
   7,500y \geq 325,000 - 175,000
   \]
   \[
   7,500y \geq 150,000
   \]

   Divide both sides by \( 7,500 \) to solve for \( y \):
   \[
   y \geq \frac{150,000}{7,500}
   \]
   \[
   y \geq 20
   \]

Thus, the inequality that describes the set of years where the landfill is at or above capacity is:
\[
y \geq 20
\]

This means that starting 20 years after January 1, 2000, which is January 1, 2020, the landfill will be at or above its capacity.

Would you like more details or have any questions? Here are some related questions for further practice:

1. What will be the amount of trash in the landfill on January 1, 2025?
2. How many years will it take for the landfill to reach 250,000 tons of trash?
3. If the annual increase in trash is reduced to 5,000 tons per year, how does the inequality change?
4. What would be the landfill's capacity if the annual increase was instead 10,000 tons?
5. Calculate the total amount of trash in the landfill 15 years after January 1, 2000.
6. If the landfill's initial amount of trash was 150,000 tons, how many years would it take to reach capacity?
7. How many years will it take to fill half of the landfill's capacity starting from 175,000 tons?
8. What is the average annual increase in the amount of trash if the landfill reaches capacity in 18 years?

**Tip:** When solving inequalities involving real-world scenarios, always interpret the final result in the context of the problem to ensure it makes sense and is feasible.

Understand how to determine the years when a landfill reaches or exceeds its capacity using inequalities. Solve the problem step-by-step with initial trash amount, capacity, and annual increase data. Suitable for high school students studying inequalities and linear functions.

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