Final answer:
The equation |4x-7| = -1 has no solution because absolute values are always non-negative, and it contradicts the definition of an absolute value as a distance from zero on the number line.
Step-by-step explanation:
The equation |4x-7| = -1 has no solution because absolute values cannot be negative, by definition. The absolute value of a number refers to its distance from zero on the number line, regardless of direction, which means it's always a non-negative number. Therefore, the expression |4x-7|, which represents the absolute value of 4x-7, must also be non-negative.
The mistaken concept that an absolute value can be negative, like expressed in the equation |4x-7| = -1, contradicts the fundamental property that an absolute value can only be zero or positive. This property is analogous to the rules of multiplication and division by which two positive or two negative numbers yield a positive product or quotient. Thus, it's impossible for |4x-7| to equal -1, and the equation has no real solution.