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The equation given in the problem is \(\lvert x - 6.4 \rvert = 83\), where \(x\) represents Sam's running time, and 6.4 seconds represents the possible variation from his average time of 83 seconds. The problem asks if this equation correctly represents the situation described.

However, there seems to be a mistake in the equation. The absolute value equation is typically used to express a range of values around a central point, but in this case, the correct equation should represent how much Sam's time \(x\) varies around the average time of 83 seconds.

### Correct Equation:
The equation should be \(\lvert x - 83 \rvert = 6.4\).

This equation means that the difference between Sam's running time \(x\) and his average time (83 seconds) can be at most 6.4 seconds, either more or less. This correctly reflects the situation described in the problem.

### Explanation of the Incorrect Equation:
The given equation \(\lvert x - 6.4 \rvert = 83\) would imply that the difference between Sam's time and 6.4 seconds is 83 seconds, which does not make sense in the context of the problem.

Would you like more details, or do you have any questions?

Here are 5 related questions you might find interesting:
1. How do you solve an absolute value equation like \(\lvert x - 83 \rvert = 6.4\)?
2. What is the range of Sam's possible running times based on the correct equation?
3. How can you graphically represent an absolute value equation?
4. How would the equation change if Sam's average time and possible variation were different?
5. Can absolute value equations be used to represent real-life scenarios other than time variations?

**Tip:** When solving absolute value equations, remember that \(\lvert a \rvert = b\) implies two possible equations: \(a = b\) and \(a = -b\).

Sam runs track and can run a 400 meter distance in an average time of 83 seconds. The fastest and slowest times he runs can vary from his average by 6.4 seconds. He wanted to find a way to express this using an equation and used the equation |x - 6.4| = 83. Which of the following explains if his equation is correct?

Explore how to use absolute value equations to represent variations in Sam's 400 meter run times. Understand why the correct equation is |x - 83| = 6.4, where x is the time, 83 seconds is the average, and 6.4 seconds is the variation. Suitable for middle school students.

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