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The equation \( |4x - 7| = -1 \) has no solution because the absolute value, \( |4x - 7| \), is always non-negative (i.e., zero or positive) for any real number \( x \). It cannot equal a negative number. Therefore, there are no values of \( x \) that can satisfy the equation where the absolute value equals -1.

Would you like to explore more about absolute value equations or have any other questions?

1. What is the definition of absolute value?
2. How would you solve \( |4x - 7| = 5 \)?
3. Can an absolute value equation have more than two solutions?
4. Why can the absolute value function never yield a negative result?
5. How can graphing \( |4x - 7| \) help understand solutions to equations involving absolute values?

Tip: Always remember that absolute values are defined to measure the "distance" of a number from zero on the number line, which means they can only yield zero or positive values.

Without calculating, how do you know that the equation |4x - 7| = -1 has no solution?

Understand why the absolute value equation |4x - 7| = -1 has no solution without any calculations. This explanation highlights key algebraic concepts related to absolute value properties and is suitable for students in Grades 7-10.

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