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Absolute value refers to the distance of a number from zero on the number line. It is always a non-negative number. The symbol for absolute value is \(...|x|...\). This means:

A positive number remains the same under absolute value. For example, \(|5| = 5...\).
A negative number becomes positive. For example, \(|-5| = 5...\).
Zero has an absolute value of zero. For example, \(|0| = 0...\).

In general, \(|x| \geq 0...\) for any real number \( x \).
Understanding absolute value is important, as it will help us compare \(|x^4|...\) and \(|x^5|...\).

Inequalities tell us about the relative size of two values. In this exercise, we are given that \(-1 < x < 1...\) but \( x \eq 0...\).
This inequality means that \( x \) is between -1 and 1, excluding zero. We break this down into two cases:

1. Case 1: \(0 < x < 1...\). Here, \( x \) is a positive number less than 1.
2. Case 2: \(-1 < x < 0...\). Here, \( x \) is a negative number greater than -1.
When dealing with inequalities, it's important to remember the rules for manipulating them. For example, if you multiply or divide both sides of an inequality by a negative number, the direction of the inequality flips.
This concept helps us understand how the values of \(|x^4|...\) and \(|x^5|...\) change relative to each other based on their powers.

Exponentiation involves raising a number to the power of another number. It is indicated using the notation \(x^n...\), where \(x \) is the base and \(n \) is the exponent. Here are some key principles:

Raising a number between \(-1 \) and \(1 \) to a higher power results in a smaller value. For instance, if \( 0 < x < 1...\), then \(x^5 < x^4...\), because each time you multiply a small fraction by itself, it gets even smaller.
For negative numbers in the range \(-1 < x < 0...\), the same principle applies as we only consider the absolute value.
In our case, since \(|x| < 1...\), raising \(x \) to higher powers makes its absolute value smaller. Specifically, \(|x^5| < |x^4|...\).
Thus, by understanding the properties of exponentiation, we confirm that \( |x^5| \) will always be less than \( |x^4| \) within our defined range of \(-1 < x < 1 \) but \( x \eq 0 \).