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Absolute value is a fundamental concept in mathematics that indicates how far a number is from zero on a number line, without considering the direction. It's always a non-negative value. We denote the absolute value of a number \( x \) as \( |x| \). If \( x \) is positive or zero, \( |x| = x \). If \( x \) is negative, \( |x| = -x \).
For example:

• If \( x = 5 \), then \( |5| = 5 \)
• If \( x = -3 \), then \( |-3| = 3 \)

In the context of our original exercise, we use the absolute value to show how the equation remains valid for both positive and negative values of \( x \). Therefore, \( |x| \) provides a unified way to express distance on the number line, simplifying solutions for various mathematical problems.

The square root is another key concept. It represents a number that, when multiplied by itself, gives the original number. For instance, the square root of 9 is 3, because \( 3 \times 3 = 9 \). We denote the square root of a number \( x \) as \( \root 2 \text of {x} \).
Importantly, the square root of \( x^2 \) is always the absolute value of \( x \), expressed as \( |x| \). This means that whether \( x \) is positive or negative, \( \root 2 \text of {x^2} = |x| \).
For our equation \( |x| = \root 2 \text of {x^2} \), this property helps us establish that the pathway from \( |x| \) to \( \root 2 \text of {x^2} \) is symmetric. Whether \( x \) is positive or negative, the outcome is the same, reinforcing that the equation holds true universally.

Real numbers include all the numbers that can be found on the number line. This comprises both rational numbers (fractions and integers) and irrational numbers (numbers that cannot be written as a simple fraction, like \( \root 2 \text of {2} \) or \( \text pi \)).
In the context of the equation \( |x| = \root 2 \text of {x^2} \), real numbers play a crucial role because \( x \) can be any real number. There's no restriction on \( x \), making the equation true for an infinite range of real numbers. This includes negative numbers, positive numbers, and zero.
Thus, because the absolute value and square root functions produce consistent results over the entire set of real numbers, the equation has infinitely many solutions. Essentially, if there are no boundaries on the values \( x \) can take within the real number system, the solutions are endless.