why is |-9| = 9 a valid answer but |9| = -9 an invalid answer

The expression \( |-9| = 9 \) is valid because the absolute value of any number is always non-negative. The absolute value function, denoted by \( |x| \), returns the distance of \( x \) from zero on the number line, without regard to its sign. Here's how it works:

1. **Absolute value of negative numbers**: 
   - For \( |-9| \), the distance of \(-9\) from zero is 9. So, \( |-9| = 9 \).
   - This is correct because the absolute value of a negative number is its positive counterpart.

2. **Absolute value of positive numbers**: 
   - For \( |9| \), the distance of 9 from zero is also 9, so \( |9| = 9 \). 
   - It would be incorrect to say \( |9| = -9 \) because absolute values cannot result in negative numbers. 

Thus, the absolute value of any number, whether negative or positive, is always a non-negative number, making \( |9| = -9 \) an invalid statement.

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Here are 5 related questions:
1. What is the absolute value of 0?
2. How does the absolute value function behave with decimals?
3. Can absolute value ever be negative?
4. How is the absolute value function used in solving equations?
5. Why does \( |x| = -y \) have no solutions if \( y > 0 \)?

**Tip**: When solving absolute value equations, always check whether the solution could be negative, as this can lead to no solution.

Explore why the absolute value of -9 equals 9 but |9| cannot be equal to -9. This problem explains the fundamental concept of absolute value and the non-negativity property, suitable for middle school students (Grades 6-8).

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