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Understanding the absolute value is crucial for solving various mathematical problems, especially when dealing with real numbers. The absolute value of a real number refers to its distance from zero on the number line, regardless of direction. This means the absolute value is always non-negative. For instance, the absolute value of both 3 and -3 is 3.
When we talk about the properties of absolute values, one of the most fundamental ones is how absolute values interact with multiplication, as seen in our exercise. The rule \( |x||y| = |xy| \) holds true for all real numbers, x and y. This is because the absolute value of a product is the product of the absolute values. So if either or both numbers are negative, their product is positive, just like their absolute values when multiplied.
Another key property is that the absolute value of a number is that number if it is non-negative, and the opposite of that number if it is negative. This property is essential when solving equations and inequalities involving absolute values and shows up in our exercise as well.

Real numbers form the backbone of mathematics, representing all possible numbers along the continuum of the number line. This includes all rationals and irrationals, whether they are positive, negative, or zero. Often when we engage in solving problems, the domain we consider is ‘all real numbers’ unless specified otherwise.
Within the realm of real numbers, we have subsets such as integers, which include whole numbers and their negatives, and natural numbers, which are the positive integers starting from 1. Understanding the different categories of real numbers can help immensely in approaching proofs or exercises, just like in the proof presented in which the cases were based upon the different possibilities for two real numbers, x and y.
By proving that certain properties hold for all real numbers, we establish that the rules we use are universally applicable within this number system, which is why it was important to show that the property \( |xy| = |x||y| \) holds no matter what kind of real numbers x and y are.

Mathematical proofs are logical arguments that establish the truth of a mathematical statement. Proofs employ a variety of methods, and proof by cases is one of these techniques. It entails dividing the proof into several cases that exhaust all possibilities. The objective is to show that the statement to be proved holds true in each individual case.
In the exercise presented, proof by cases was used to demonstrate a fundamental property of absolute values over real numbers. By considering all the different signs that x and y could take on (positive, negative, or zero), and showing that \( |x y| = |x||y| \) holds in every scenario, the statement was proven to be universally true. This approach is particularly useful when dealing with piecewise definitions or properties like absolute value, where behavior changes based on input conditions.