Prove: X = X_+ - X_- And |x| = X_+ + X_-

Hey guys! Today, we're diving into a cool little problem from the world of real numbers. We've got these special notations, x_+ and x_-, defined in terms of x and its absolute value |x|. Our mission? To prove that x can be expressed as the difference between x_+ and x_-, and that the absolute value of x is the sum of x_+ and x_-. Buckle up, because we're about to embark on a mathematical adventure that's both enlightening and fun!

Understanding the Definitions

Before we jump into the proofs, let's make sure we're all on the same page with the definitions. For any real number x, we have:

• x_+ = (x + |x|) / 2
• x_- = (-x + |x|) / 2

These definitions might seem a bit abstract at first, but they're actually quite intuitive. Think of x_+ as the "positive part" of x and x_- as the "negative part" of x. Let's break down these definitions further to truly grasp what they mean.

Delving Deeper into x_+

The expression x_+ = (x + |x|) / 2 essentially captures the non-negative component of x. Consider two scenarios:

1. When x is non-negative (x ≥ 0): In this case, |x| = x, so x_+ = (x + x) / 2 = x. This means that when x is already non-negative, x_+ simply equals x.
2. When x is negative (x < 0): Here, |x| = -x, and thus x_+ = (x + (-x)) / 2 = 0. This shows that when x is negative, x_+ becomes zero, effectively discarding the negative value.

So, x_+ acts as a filter, preserving x when it's non-negative and zeroing it out when it's negative. This behavior is crucial in various mathematical contexts, especially when dealing with inequalities and piecewise functions. Understanding this nuance is key to mastering more complex concepts later on.

Unpacking the Meaning of x_-

Now, let's turn our attention to x_- = (-x + |x|) / 2, which represents the non-negative component of -x. Again, let's analyze two cases:

1. When x is non-negative (x ≥ 0): In this scenario, |x| = x, leading to x_- = (-x + x) / 2 = 0. Consequently, when x is non-negative, x_- becomes zero, eliminating any positive value.
2. When x is negative (x < 0): In this situation, |x| = -x, so x_- = (-x + (-x)) / 2 = -x. This implies that when x is negative, x_- equals the negation of x, effectively turning the negative value into a positive one.

Thus, x_- serves as a filter that preserves the negation of x when x is negative and zeros it out when x is non-negative. This behavior complements x_+, allowing us to isolate and manipulate the negative component of x. This is especially useful in scenarios where we need to deal with absolute values or piecewise functions.

Proving x = x_+ - x_-

Alright, now that we've got a handle on what x_+ and x_- mean, let's prove the first part: x = x_+ - x_-. To do this, we'll substitute the definitions of x_+ and x_- into the right-hand side of the equation and see if we can simplify it to x.

So, we have:

x_+ - x_- = [(x + |x|) / 2] - [(-x + |x|) / 2]

Combining the fractions, we get:

x_+ - x_- = (x + |x| + x - |x|) / 2

Notice that the |x| terms cancel each other out, leaving us with:

x_+ - x_- = (2x) / 2

And finally:

x_+ - x_- = x

Boom! We've proven that x = x_+ - x_-. This result tells us that any real number x can be expressed as the difference between its "positive part" (x_+) and its "negative part" (x_-). This is a fundamental concept that's used in many areas of mathematics.

Proving |x| = x_+ + x_-

Now, let's move on to the second part of our mission: proving that |x| = x_+ + x_-. Again, we'll substitute the definitions of x_+ and x_- into the right-hand side of the equation and simplify.

We have:

x_+ + x_- = [(x + |x|) / 2] + [(-x + |x|) / 2]

Combining the fractions, we get:

x_+ + x_- = (x + |x| - x + |x|) / 2

This time, the x terms cancel each other out, leaving us with:

x_+ + x_- = (2|x|) / 2

And finally:

x_+ + x_- = |x|

There you have it! We've proven that |x| = x_+ + x_-. This result shows that the absolute value of x is the sum of its "positive part" and its "negative part". This makes intuitive sense because the absolute value of a number is always non-negative, and x_+ and x_- are defined to be non-negative.

Wrapping Up

So, what have we learned today? We've successfully proven two important properties of real numbers using the notations x_+ and x_-. These properties are:

• x = x_+ - x_-
• |x| = x_+ + x_-

These results might seem simple, but they're actually quite powerful. They provide us with a way to decompose any real number into its positive and negative components, which can be useful in many different contexts. From solving equations to analyzing functions, these properties can help us simplify complex problems and gain a deeper understanding of the mathematical world. Keep these formulas in mind, guys, they might just come in handy someday! Remember, mathematics isn't just about memorizing formulas; it's about understanding the underlying concepts and applying them creatively.