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Absolute value represents the distance of a number from 0 on the number line, regardless of its direction. For example, the absolute value of both 12 and -12 is 12 because both are 12 units away from 0. Mathematically, we denote the absolute value of a number 'a' as \(|a|\).

It's important to remember that the absolute value is always a non-negative number. Whether the original number is positive or negative, the absolute value strips away the sign and considers only the numerical part. This concept is key in various mathematical operations and comparisons, such as the ones in the example exercise.

Negative numbers are numbers less than zero, expressed with a minus sign (-). They indicate a value that is below zero on the number line. For instance, -5 is 5 units to the left of 0.

When dealing with negative numbers, it’s essential to understand that their relative size follows the opposite rule compared to positive numbers. For example, -3 is greater than -5, even though 3 is less than 5 in positive terms. This reversal can sometimes be confusing but is crucial when comparing negative numbers.

In the provided exercise, after calculating the absolute values, we negate them, which essentially shifts our comparison into the realm of negative numbers. Hence, -12 is greater than -15.

Inequalities are expressions that show the relationship between two values when they are not equal. Common symbols used for inequalities include > (greater than), < (less than), ≥ (greater than or equal to), and ≤ (less than or equal to).

Comparing negative numbers involves inequalities. Consider the exercise example: after negating the absolute values, we have -12 and -15. Using the inequality symbol, we observe that -12 is greater than -15.

Understanding how to correctly use inequalities when dealing with absolute values and negative numbers is fundamental in many areas of mathematics. It helps in solving equations, analyzing functions, and making logical conclusions based on numerical data.