For $0 \ \textless \ X \ \textless \ 2$, What Is ∣ X 2 − 2 X ∣ X − 2 \frac{\left|x^2 - 2x\right|}{x - 2} X − 2 ∣ X 2 − 2 X ∣ ​ ?

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Introduction

When dealing with absolute value expressions, it's essential to understand the properties and behavior of absolute value functions. In this article, we'll delve into the world of absolute value expressions and explore the solution to the given problem: $\frac{\left|x^2 - 2x\right|}{x - 2}$ for $0 < x < 2$. We'll break down the problem into manageable steps, using mathematical concepts and techniques to arrive at the final solution.

Understanding Absolute Value Functions

Before we dive into the problem, let's take a moment to understand the concept of absolute value functions. The absolute value function, denoted by $|x|$, is defined as:

$$|x| = \begin{cases} x, & \text{if } x \geq 0 \\ -x, & \text{if } x < 0 \end{cases}$$

This function takes any real number $x$ and returns its distance from zero on the number line. In other words, it returns the magnitude of $x$, without considering its sign.

Evaluating the Expression

Now that we have a solid understanding of absolute value functions, let's focus on the given expression: $\frac{\left|x^2 - 2x\right|}{x - 2}$. To evaluate this expression, we need to consider the properties of absolute value functions and the behavior of the expression within the given interval $0 < x < 2$.

Case 1: $x^2 - 2x \geq 0$

When $x^2 - 2x \geq 0$, the absolute value expression simplifies to:

$$\left|x^2 - 2x\right| = x^2 - 2x$$

Substituting this into the original expression, we get:

$$\frac{\left|x^2 - 2x\right|}{x - 2} = \frac{x^2 - 2x}{x - 2}$$

Simplifying the Expression

To simplify the expression, we can factor the numerator:

$$\frac{x^2 - 2x}{x - 2} = \frac{x(x - 2)}{x - 2}$$

Canceling Common Factors

Now that we have factored the numerator, we can cancel out the common factor $(x - 2)$:

$$\frac{x(x - 2)}{x - 2} = x$$

Case 2: $x^2 - 2x < 0$

When $x^2 - 2x < 0$, the absolute value expression simplifies to:

$$\left|x^2 - 2x\right| = -(x^2 - 2x)$$

Substituting this into the original expression, we get:

$$\frac{\left|x^2 - 2x\right|}{x - 2} = \frac{-(x^2 - 2x)}{x - 2}$$

Simplifying the Expression

To simplify the expression, we can factor the numerator:

$$\frac{-(x^2 - 2x)}{x - 2} = \frac{-x(x - 2)}{x - 2}$$

Canceling Common Factors

Now that we have factored the numerator, we can cancel out the common factor $(x - 2)$:

$$\frac{-x(x - 2)}{x - 2} = -x$$

Combining the Results

Now that we have evaluated the expression for both cases, we can combine the results:

$$\frac{\left|x^2 - 2x\right|}{x - 2} = \begin{cases} x, & \text{if } x^2 - 2x \geq 0 \\ -x, & \text{if } x^2 - 2x < 0 \end{cases}$$

Conclusion

In this article, we've explored the solution to the given problem: $\frac{\left|x^2 - 2x\right|}{x - 2}$ for $0 < x < 2$. We've broken down the problem into manageable steps, using mathematical concepts and techniques to arrive at the final solution. By understanding the properties of absolute value functions and the behavior of the expression within the given interval, we've been able to simplify the expression and arrive at the final result.

Final Answer

The final answer to the problem is:

$$\frac{\left|x^2 - 2x\right|}{x - 2} = \begin{cases} x, & \text{if } x^2 - 2x \geq 0 \\ -x, & \text{if } x^2 - 2x < 0 \end{cases}$$

Note that this answer is valid for the given interval $0 < x < 2$.

$$

Q: What is the absolute value function?

A: The absolute value function, denoted by $|x|$, is defined as:

$$|x| = \begin{cases} x, & \text{if } x \geq 0 \\ -x, & \text{if } x < 0 \end{cases}$$

This function takes any real number $x$ and returns its distance from zero on the number line.

Q: How do I evaluate the expression $\frac{\left|x^2 - 2x\right|}{x - 2}$?

A: To evaluate the expression, you need to consider the properties of absolute value functions and the behavior of the expression within the given interval $0 < x < 2$. You can break down the problem into two cases: when $x^2 - 2x \geq 0$ and when $x^2 - 2x < 0$.

Q: What happens when $x^2 - 2x \geq 0$?

A: When $x^2 - 2x \geq 0$, the absolute value expression simplifies to:

$$\left|x^2 - 2x\right| = x^2 - 2x$$

Substituting this into the original expression, you get:

$$\frac{\left|x^2 - 2x\right|}{x - 2} = \frac{x^2 - 2x}{x - 2}$$

Q: What happens when $x^2 - 2x < 0$?

A: When $x^2 - 2x < 0$, the absolute value expression simplifies to:

$$\left|x^2 - 2x\right| = -(x^2 - 2x)$$

Substituting this into the original expression, you get:

$$\frac{\left|x^2 - 2x\right|}{x - 2} = \frac{-(x^2 - 2x)}{x - 2}$$

Q: How do I simplify the expression $\frac{x^2 - 2x}{x - 2}$?

A: To simplify the expression, you can factor the numerator:

$$\frac{x^2 - 2x}{x - 2} = \frac{x(x - 2)}{x - 2}$$

Q: How do I simplify the expression $\frac{-(x^2 - 2x)}{x - 2}$?

A: To simplify the expression, you can factor the numerator:

$$\frac{-(x^2 - 2x)}{x - 2} = \frac{-x(x - 2)}{x - 2}$$

Q: What is the final answer to the problem?

A: The final answer to the problem is:

$$\frac{\left|x^2 - 2x\right|}{x - 2} = \begin{cases} x, & \text{if } x^2 - 2x \geq 0 \\ -x, & \text{if } x^2 - 2x < 0 \end{cases}$$

Note that this answer is valid for the given interval $0 < x < 2$.

Q: What is the significance of the absolute value function in this problem?

A: The absolute value function plays a crucial role in this problem by allowing us to simplify the expression and arrive at the final result. By understanding the properties of absolute value functions, we can break down the problem into manageable steps and arrive at the final answer.

Q: How can I apply the concepts learned in this problem to other mathematical problems?

A: The concepts learned in this problem can be applied to other mathematical problems that involve absolute value functions and expressions. By understanding the properties of absolute value functions and how to simplify expressions, you can tackle a wide range of mathematical problems with confidence.

Q: What are some common mistakes to avoid when working with absolute value functions?

A: Some common mistakes to avoid when working with absolute value functions include:

• Not considering the properties of absolute value functions
• Not breaking down the problem into manageable steps
• Not simplifying the expression correctly
• Not considering the behavior of the expression within the given interval

By avoiding these common mistakes, you can ensure that you arrive at the correct solution to the problem.