The given equation is:

$|x - 6| = -1$

Analysis:

The absolute value function, $|x - 6|$, represents the distance of $x - 6$ from 0, and it is always non-negative. This means that:

$|x - 6| \geq 0 \quad \text{for all } x.$

Thus, it is impossible for $|x - 6|$ to equal $-1$, as the absolute value can never be negative.

Conclusion:

The equation has no solution.

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Would you like additional details or examples about absolute value equations?

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Here are 5 related questions for further exploration:

1. What are the properties of absolute value functions?
2. How do you solve equations where the absolute value equals a positive number (e.g., $|x - 3| = 5$)?
3. How do absolute value inequalities (e.g., $|x - 2| > 3$) differ from equations?
4. What is the graphical representation of absolute value equations and inequalities?
5. How does squaring both sides of an equation affect absolute value equations?

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Tip: Always check if an absolute value equation is valid before solving. If the absolute value is set equal to a negative number, the equation has no solution.