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Real numbers include every number that can be found on the number line. This vast category encompasses all the numbers we typically use, such as positive numbers, negative numbers, zero, and all the fractions and decimals in between. In mathematical terms, the real numbers are composed of rational numbers (like \frac{1}{2} or \frac{4}{3}) and irrational numbers (such as \( \sqrt{2} \) or \( \pi \) which cannot be expressed as a simple fraction).

When discussing inequalities that involve real numbers, the main principle to understand is that these inequalities reflect the ordering of numbers on the number line. For example, \( 3 \leq 5 \) means 3 is to the left of 5 on the number line. Similarly, when we look at the absolute value inequality in the original exercise, it’s a statement about the location of every real number \( x \) in relation to its absolute value on the number line.

Proving an inequality often requires demonstrating that the statement is true for all possible scenarios. The exercise provided challenges us to prove that \( -|x| \leq x \leq |x| \) for all real numbers. To tackle this proof, we employed a case analysis strategy, which is a common technique in proofs involving absolute values.

This approach involves splitting the problem into distinct cases based on the definition of the absolute value. Namely, if the number is non-negative (including zero), then the absolute value is simply the number itself. On the other hand, if the number is negative, the absolute value is the number's opposite. By verifying the given inequality for both cases, we confirmed its validity across the entire set of real numbers. This methodical process helps establish the truth of the inequality in a clear and logical manner.

Mathematical logic is the foundation that supports the reasoning and structured approach used in solving problems like the absolute value inequality we discussed. Logic helps dictate how we approach different cases within a proof and arrive at a valid conclusion.

In our solution, we utilized logical operators such as 'and' (\( \wedge \)) and 'or' (\( \vee \)) implicitly in deciding that if \( x \) is non-negative or negative, different properties would apply. Through logical analysis, we determined that for non-negative \( x \)s, the proposition \( -x \leq x \leq x \) is true; similarly, for negative \( x \)s, \( x \leq x \leq -x \) holds true. By covering all possible real numbers in this manner, we used logical deduction to verify the inequality for all cases, thus providing a proof of the original statement.