A tank starts off containing 1,260 gallons of water, and 45 gallons of water leaks from the tank every hour.  At the same time, an empty pool is filled with water at a rate of 580 gallons per hour.  

Which inequality represents all possible values of x, the number of hours that have passed when the tank contains less water than the pool?

We need to set up an inequality that describes the point in time when the amount of water in the tank becomes less than the amount of water in the pool. Let's break it down:

### Tank:
- Initially, the tank contains 1,260 gallons.
- Every hour, 45 gallons leaks from the tank.
- After \(x\) hours, the amount of water in the tank is:
  \[
  \text{Water in the tank} = 1260 - 45x
  \]

### Pool:
- Initially, the pool is empty.
- Every hour, 580 gallons of water is added to the pool.
- After \(x\) hours, the amount of water in the pool is:
  \[
  \text{Water in the pool} = 580x
  \]

### Inequality:
We want to find when the water in the tank is **less than** the water in the pool, so the inequality is:
\[
1260 - 45x < 580x
\]

### Solving the inequality:
1. Move all terms involving \(x\) to one side:
   \[
   1260 < 580x + 45x
   \]
   \[
   1260 < 625x
   \]

2. Solve for \(x\):
   \[
   x > \frac{1260}{625}
   \]
   \[
   x > 2.016
   \]

Since time is typically measured in whole hours, we round up to the nearest whole number. Therefore, the tank will contain less water than the pool after **3 hours**.

### Final inequality:
\[
x > 2.016 \quad \text{(or simply \(x \geq 3\) in practical terms)}
\]

Would you like further details on solving this inequality or any other questions?

#### 5 Related Questions:
1. How do you solve inequalities with variables on both sides?
2. What are the differences between strict inequalities (<) and non-strict inequalities (≤)?
3. How do you interpret word problems to set up equations or inequalities?
4. How can you graph inequalities on a number line?
5. How do you round decimals in real-world contexts like this one?

#### Tip:
Always check units and assumptions (e.g., rounding) when dealing with real-world problems to ensure the solution makes sense in context.

A tank starts off containing 1,260 gallons of water, and 45 gallons of water leaks from the tank every hour. At the same time, an empty pool is filled with water at a rate of 580 gallons per hour. Which inequality represents all possible values of x, the number of hours that have passed when the tank contains less water than the pool?

Learn how to solve a real-world word problem involving a leaking water tank and a filling pool. This detailed solution demonstrates setting up and solving a linear inequality step by step, covering algebraic concepts such as linear equations and inequalities. Suitable for students in Grades 6-8.

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