91Ó°ÊÓ

In this exercise, we deal with nonzero real numbers. A real number is any value that can represent a distance along a number line, including both positive and negative numbers, as well as zero. However, when we specify 'nonzero', we exclude zero from our range. That’s because we can’t divide by zero—such an operation is undefined in mathematics.
In our problem, we are asked to start with an inequality involving a nonzero real number, specifically: \((x - \frac{1}{x})^2 \geq 0\). To understand why this is always true for nonzero \(x\), recall that the square of any real number (positive or negative) is always non-negative. Therefore, this inequality will hold for any nonzero value of \(x\).

Algebraic manipulation is the process of rearranging and simplifying expressions to get a desired form or solution. Here's how we apply that to our given problem:

• Starting with the inequality: \((x - \frac{1}{x})^2 \geq 0\)
• We expand the left side: \((x - \frac{1}{x})^2 = x^2 - 2 \cdot x \cdot \frac{1}{x} + (\frac{1}{x})^2 \)

After simplifying, we get: \x^2 - 2 + \frac{1}{x^2} \geq 0\. Now, to isolate the desired expression, we add 2 to both sides of the inequality:
\ x^2 - 2 + \frac{1}{x^2} + 2 \geq 0 + 2 \ which simplifies to: \ x^2 + \frac{1}{x^2} \geq 2 \.
This step-by-step algebraic manipulation helps us derive the required inequality from the original one.

Inequations, often referred to as inequalities, are mathematical statements that are used to compare the sizes of two values or expressions. Unlike equations, which state that two expressions are equal, inequations indicate that quantities are not necessarily equal but have a specific order relation (greater than, less than, etc.).
In our example, the original inequation \((x - \frac{1}{x})^2 \geq 0\) uses the 'greater than or equal to' relation to demonstrate that the square of any real number is non-negative. This is fundamental because it allows us to safely perform algebraic manipulations to derive the conclusion \x^2 + \frac{1}{x^2} \geq 2\. By understanding and applying properties of real numbers and inequalities, we can demonstrate this important mathematical truth effectively.