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In mathematics, the absolute value of a number is essentially its distance from zero on the number line, regardless of direction. Think of it as the 'numerical value' without any concern whether it is positive or negative.

When comparing the absolute value of two numbers, you temporarily disregard the signs. For example, the absolute value of both \(-27\) and \(-39\) is represented as \(|-27| = 27\) and \(|-39| = 39\), respectively. This means we are only considering how far each number is from zero.

It’s important to note that the absolute value is always a non-negative number. Even if the original number is negative, the absolute value is positive. In the given exercise, the absolute values help us compare the size of two negative numbers disregarding their sign.

Inequalities are a way of expressing the relative size or order of two numbers. The symbols used in inequalities are \(<\) for 'less than' and \(>\) for 'greater than'. Understanding how to interpret these symbols is crucial when comparing values.

When two negative numbers are put into an inequality, it may seem counterintuitive since a larger negative number is actually closer to zero and therefore 'less negative'. For example, \(-2\) is greater than \(-3\), as \(-2\) is closer to zero. In the exercise, we concluded that since \(-27\) is closer to zero than \(-39\), it is the greater number in terms of its position on the number line, hence the correct symbol between \(-27\) and \(-39\) is \(>\).

When performing arithmetic operations with negative numbers, the standard rules of arithmetic are slightly modified. One key concept is that when comparing negative numbers, the number with the larger magnitude (absolute value) is the lesser number.

In the context of the exercise, it was necessary to remember that the number with the bigger magnitude, when negative, is actually 'less than' in terms of its value, despite appearing larger when looking at the digits alone. This is why \(-39\) is less than \(-27\), even though 39 is larger than 27. To perform operations correctly, keeping this rule in mind is essential to ensure that the arithmetic of negative numbers is accurate and follows the established conventions of mathematics.