Which Of The Following Is Equal To The Rational Expression When $x \neq 2$ Or $-4$?$\frac{5(x-2)}{(x-2)(x+4)}$A. $\frac{5}{x-2}$ B. $\frac{1}{x-4}$ C. $\frac{5}{x+4}$ D. $\frac{1}{x-2}$

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Introduction

Rational expressions are a fundamental concept in algebra, and simplifying them is a crucial skill to master. In this article, we will explore the process of simplifying rational expressions, with a focus on the given problem: 5(x−2)(x−2)(x+4)\frac{5(x-2)}{(x-2)(x+4)}. We will examine the options provided and determine which one is equal to the rational expression when x≠2x \neq 2 or −4-4.

Understanding Rational Expressions

A rational expression is a fraction that contains variables and/or constants in the numerator and/or denominator. Rational expressions can be simplified by canceling out common factors between the numerator and denominator. However, it's essential to note that rational expressions can be undefined when the denominator is equal to zero.

Simplifying the Given Rational Expression

To simplify the given rational expression, we need to factor the numerator and denominator. The numerator can be factored as 5(x−2)5(x-2), and the denominator can be factored as (x−2)(x+4)(x-2)(x+4). We can now cancel out the common factor of (x−2)(x-2) between the numerator and denominator.

\frac{5(x-2)}{(x-2)(x+4)} = \frac{5}{x+4}

Analyzing the Options

Now that we have simplified the rational expression, let's examine the options provided:

A. 5x−2\frac{5}{x-2}

B. 1x−4\frac{1}{x-4}

C. 5x+4\frac{5}{x+4}

D. 1x−2\frac{1}{x-2}

Option A: 5x−2\frac{5}{x-2}

Option A is not equal to the simplified rational expression. The denominator of option A is x−2x-2, which is not the same as the denominator of the simplified rational expression, x+4x+4.

Option B: 1x−4\frac{1}{x-4}

Option B is not equal to the simplified rational expression. The denominator of option B is x−4x-4, which is not the same as the denominator of the simplified rational expression, x+4x+4.

Option C: 5x+4\frac{5}{x+4}

Option C is equal to the simplified rational expression. The denominator of option C is x+4x+4, which is the same as the denominator of the simplified rational expression.

Option D: 1x−2\frac{1}{x-2}

Option D is not equal to the simplified rational expression. The denominator of option D is x−2x-2, which is not the same as the denominator of the simplified rational expression, x+4x+4.

Conclusion

In conclusion, the correct answer is option C: 5x+4\frac{5}{x+4}. This is equal to the simplified rational expression when x≠2x \neq 2 or −4-4. It's essential to remember that rational expressions can be undefined when the denominator is equal to zero, and we must always check for this condition when simplifying or evaluating rational expressions.

Final Thoughts

Q: What is a rational expression?

A: A rational expression is a fraction that contains variables and/or constants in the numerator and/or denominator.

Q: Why is it essential to simplify rational expressions?

A: Simplifying rational expressions is crucial because it helps to:

  • Reduce the complexity of the expression
  • Make it easier to evaluate and solve equations
  • Identify any undefined conditions
  • Improve the overall understanding of the underlying concepts

Q: How do I simplify a rational expression?

A: To simplify a rational expression, follow these steps:

  1. Factor the numerator and denominator
  2. Cancel out any common factors between the numerator and denominator
  3. Check for any undefined conditions (i.e., when the denominator is equal to zero)

Q: What is the difference between a rational expression and a rational number?

A: A rational number is a number that can be expressed as the ratio of two integers, i.e., a fraction. A rational expression, on the other hand, is a fraction that contains variables and/or constants in the numerator and/or denominator.

Q: Can I simplify a rational expression with a variable in the denominator?

A: Yes, you can simplify a rational expression with a variable in the denominator. However, you must be careful to check for any undefined conditions, i.e., when the denominator is equal to zero.

Q: How do I know when a rational expression is undefined?

A: A rational expression is undefined when the denominator is equal to zero. To check for this condition, set the denominator equal to zero and solve for the variable.

Q: Can I simplify a rational expression with a negative exponent?

A: Yes, you can simplify a rational expression with a negative exponent. To do this, rewrite the expression with a positive exponent and then simplify.

Q: How do I simplify a rational expression with a fraction in the numerator or denominator?

A: To simplify a rational expression with a fraction in the numerator or denominator, follow these steps:

  1. Simplify the fraction in the numerator or denominator
  2. Factor the numerator and denominator
  3. Cancel out any common factors between the numerator and denominator
  4. Check for any undefined conditions

Q: Can I simplify a rational expression with a radical in the numerator or denominator?

A: Yes, you can simplify a rational expression with a radical in the numerator or denominator. To do this, follow these steps:

  1. Simplify the radical in the numerator or denominator
  2. Factor the numerator and denominator
  3. Cancel out any common factors between the numerator and denominator
  4. Check for any undefined conditions

Q: How do I know when to use a rational expression versus a rational number?

A: Use a rational expression when you need to represent a relationship between variables or constants, and use a rational number when you need to represent a specific value.

Q: Can I simplify a rational expression with a complex fraction?

A: Yes, you can simplify a rational expression with a complex fraction. To do this, follow these steps:

  1. Simplify the complex fraction
  2. Factor the numerator and denominator
  3. Cancel out any common factors between the numerator and denominator
  4. Check for any undefined conditions

Conclusion

Simplifying rational expressions is a crucial skill in algebra, and it requires a deep understanding of the underlying concepts. By following the steps outlined in this article, you can simplify rational expressions with ease and confidence. Remember to always factor the numerator and denominator, cancel out common factors, and check for undefined conditions. With practice and patience, you will become proficient in simplifying rational expressions and tackling even the most challenging problems.