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Absolute value is a concept in mathematics that measures how far a number is from zero on a number line. The absolute value of a number is denoted by putting the number in vertical bars, for example, \(|a|\). Absolute values have special properties which are vital for solving equations:

• They are always non-negative, which means \(|a| \geq 0\) for any real number \(a\).
• The absolute value of zero is zero, meaning \(|0| = 0\).
• For any real number \(a\), both \(|a| = a\) if \(a\) is positive and \(|a| = -a\) if \(a\) is negative.
  
  Understanding these properties is crucial when solving absolute value equations because they determine that no absolute value expression can equal a negative number, such as -5 in the given problem. This fundamental concept helps discern whether an equation has valid solutions in the real numbers.

Non-negative numbers are numbers that are either zero or positive. This includes the entire set of positive numbers as well as zero itself. In mathematical terms, a number \(n\) is non-negative if \(n \geq 0\).

Absolute values are exclusively non-negative by definition, never resulting in a negative figure. This rule comes from the understanding that absolute value measures distance, and distances cannot be negative. So, when we say \(|x-4| = -5\), we face a contradiction. Because the absolute value cannot be less than zero, a negative result indicates a misunderstanding or impossibility in the problem context.

Recognizing the nature of non-negative numbers is key to evaluating whether equations involving absolute values can have solutions.

Real numbers are all the numbers that can be found on the number line, including both positive and negative numbers, zero, integers, fractions, and irrational numbers. When solving equations, identifying solutions within real numbers means finding any numeric value that satisfies the equation.

In the case of the equation \(|x-4| = -5\), you check if any real number \(x\) can satisfy this equation. Given the properties of absolute values being non-negative, no real number can produce a negative absolute value. Therefore, the equation cannot have a real number solution because no \(x\) exists that makes \(|x-4|\) negative.

This conclusion is essential when solving any mathematical problems involving absolute values and ensures clear understanding through correct application of mathematical rules.