Simplify The Expression$ \frac{\frac{4}{x-1}-\frac{2}{x+1}}{\frac{x}{x-1}+\frac{1}{x+1}} }$into A Single Fraction.The Numerator Of Your Answer Is { \square$ $ The Denominator Of Your Answer Is: { \square$}$

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Introduction

When dealing with complex fractions, simplifying them can be a daunting task. However, with the right approach and techniques, it's possible to break down these expressions into manageable parts and ultimately simplify them into a single fraction. In this article, we'll focus on simplifying the given expression, which involves combining fractions in the numerator and denominator.

Understanding the Expression

The given expression is:

4x−1−2x+1xx−1+1x+1\frac{\frac{4}{x-1}-\frac{2}{x+1}}{\frac{x}{x-1}+\frac{1}{x+1}}

Our goal is to simplify this expression into a single fraction. To do this, we'll need to combine the fractions in the numerator and denominator separately.

Combining Fractions in the Numerator

To combine the fractions in the numerator, we need to find a common denominator. The common denominator for the fractions 4x−1\frac{4}{x-1} and 2x+1\frac{2}{x+1} is (x−1)(x+1)(x-1)(x+1). We can rewrite each fraction with this common denominator:

4x−1=4(x+1)(x−1)(x+1)\frac{4}{x-1} = \frac{4(x+1)}{(x-1)(x+1)}

2x+1=2(x−1)(x−1)(x+1)\frac{2}{x+1} = \frac{2(x-1)}{(x-1)(x+1)}

Now, we can subtract the two fractions:

4(x+1)(x−1)(x+1)−2(x−1)(x−1)(x+1)\frac{4(x+1)}{(x-1)(x+1)} - \frac{2(x-1)}{(x-1)(x+1)}

=4(x+1)−2(x−1)(x−1)(x+1)= \frac{4(x+1) - 2(x-1)}{(x-1)(x+1)}

=4x+4−2x+2(x−1)(x+1)= \frac{4x + 4 - 2x + 2}{(x-1)(x+1)}

=2x+6(x−1)(x+1)= \frac{2x + 6}{(x-1)(x+1)}

Combining Fractions in the Denominator

To combine the fractions in the denominator, we need to find a common denominator. The common denominator for the fractions xx−1\frac{x}{x-1} and 1x+1\frac{1}{x+1} is (x−1)(x+1)(x-1)(x+1). We can rewrite each fraction with this common denominator:

xx−1=x(x+1)(x−1)(x+1)\frac{x}{x-1} = \frac{x(x+1)}{(x-1)(x+1)}

1x+1=1(x−1)(x−1)(x+1)\frac{1}{x+1} = \frac{1(x-1)}{(x-1)(x+1)}

Now, we can add the two fractions:

x(x+1)(x−1)(x+1)+1(x−1)(x−1)(x+1)\frac{x(x+1)}{(x-1)(x+1)} + \frac{1(x-1)}{(x-1)(x+1)}

=x(x+1)+1(x−1)(x−1)(x+1)= \frac{x(x+1) + 1(x-1)}{(x-1)(x+1)}

=x2+x+x−1(x−1)(x+1)= \frac{x^2 + x + x - 1}{(x-1)(x+1)}

=x2+2x−1(x−1)(x+1)= \frac{x^2 + 2x - 1}{(x-1)(x+1)}

Simplifying the Expression

Now that we've combined the fractions in the numerator and denominator, we can rewrite the original expression as:

2x+6(x−1)(x+1)x2+2x−1(x−1)(x+1)\frac{\frac{2x + 6}{(x-1)(x+1)}}{\frac{x^2 + 2x - 1}{(x-1)(x+1)}}

To simplify this expression, we can cancel out the common factors in the numerator and denominator. The common factors are (x−1)(x-1) and (x+1)(x+1), which appear in both the numerator and denominator. Canceling these factors, we get:

2x+6x2+2x−1\frac{2x + 6}{x^2 + 2x - 1}

Conclusion

In this article, we simplified the given expression by combining fractions in the numerator and denominator. We found the common denominator for the fractions in the numerator and denominator, rewrote each fraction with the common denominator, and then combined the fractions. Finally, we simplified the expression by canceling out the common factors in the numerator and denominator. The simplified expression is:

2x+6x2+2x−1\frac{2x + 6}{x^2 + 2x - 1}

This expression can be further simplified by factoring the denominator:

2x+6(x+1)(x−1)\frac{2x + 6}{(x+1)(x-1)}

However, this is a topic for another article.

Final Answer

The numerator of the simplified expression is: 2x + 6

The denominator of the simplified expression is: (x+1)(x-1)

Introduction

In our previous article, we simplified the given expression by combining fractions in the numerator and denominator. We found the common denominator for the fractions in the numerator and denominator, rewrote each fraction with the common denominator, and then combined the fractions. Finally, we simplified the expression by canceling out the common factors in the numerator and denominator. In this article, we'll answer some frequently asked questions about simplifying expressions with fractions.

Q&A

Q: What is the first step in simplifying an expression with fractions?

A: The first step in simplifying an expression with fractions is to find the common denominator for the fractions in the numerator and denominator.

Q: How do I find the common denominator for two fractions?

A: To find the common denominator for two fractions, you need to multiply the denominators of the two fractions together. For example, if you have two fractions with denominators x-1 and x+1, the common denominator would be (x-1)(x+1).

Q: What if the fractions in the numerator and denominator have different denominators?

A: If the fractions in the numerator and denominator have different denominators, you need to find the least common multiple (LCM) of the two denominators. The LCM is the smallest number that both denominators can divide into evenly.

Q: How do I rewrite each fraction with the common denominator?

A: To rewrite each fraction with the common denominator, you need to multiply the numerator and denominator of each fraction by the necessary factors to get the common denominator. For example, if you have a fraction with a denominator of x-1 and you want to rewrite it with a denominator of (x-1)(x+1), you would multiply the numerator and denominator by (x+1).

Q: What if I have a fraction with a variable in the denominator?

A: If you have a fraction with a variable in the denominator, you need to be careful when simplifying the expression. You may need to use algebraic techniques, such as factoring or canceling out common factors, to simplify the expression.

Q: Can I simplify an expression with fractions by canceling out common factors?

A: Yes, you can simplify an expression with fractions by canceling out common factors. However, you need to be careful not to cancel out any factors that are not common to both the numerator and denominator.

Q: What if I have a fraction with a negative exponent?

A: If you have a fraction with a negative exponent, you need to be careful when simplifying the expression. You may need to use algebraic techniques, such as rewriting the fraction with a positive exponent or canceling out common factors, to simplify the expression.

Q: Can I simplify an expression with fractions by using a calculator?

A: Yes, you can simplify an expression with fractions by using a calculator. However, you need to be careful when using a calculator to simplify an expression with fractions, as it may not always give you the simplest form of the expression.

Conclusion

In this article, we answered some frequently asked questions about simplifying expressions with fractions. We discussed the first step in simplifying an expression with fractions, finding the common denominator, rewriting each fraction with the common denominator, and simplifying the expression by canceling out common factors. We also discussed some common pitfalls to avoid when simplifying expressions with fractions, such as canceling out factors that are not common to both the numerator and denominator.

Final Tips

  • Always find the common denominator for the fractions in the numerator and denominator before simplifying the expression.
  • Be careful when rewriting each fraction with the common denominator, as you may need to multiply the numerator and denominator by the necessary factors.
  • Simplify the expression by canceling out common factors, but be careful not to cancel out any factors that are not common to both the numerator and denominator.
  • Use algebraic techniques, such as factoring or canceling out common factors, to simplify expressions with fractions.
  • Be careful when using a calculator to simplify an expression with fractions, as it may not always give you the simplest form of the expression.

Additional Resources

  • For more information on simplifying expressions with fractions, see our previous article on the topic.
  • For more information on algebraic techniques for simplifying expressions with fractions, see our article on factoring and canceling out common factors.
  • For more information on using calculators to simplify expressions with fractions, see our article on calculator techniques for simplifying expressions with fractions.