Simplify The Expression:$\frac{3}{x+1} + \frac{x}{x-1}$

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Introduction

In mathematics, simplifying expressions is a crucial skill that helps us solve problems efficiently and accurately. When dealing with fractions, combining them can be a bit challenging, but with the right approach, we can simplify even the most complex expressions. In this article, we will focus on simplifying the expression $\frac{3}{x+1} + \frac{x}{x-1}$ using various techniques.

Understanding the Expression

The given expression is a sum of two fractions: $\frac{3}{x+1}$ and $\frac{x}{x-1}$. To simplify this expression, we need to find a common denominator, which is the product of the two denominators. In this case, the common denominator is $(x+1)(x-1)$.

Finding the Common Denominator

To find the common denominator, we multiply the two denominators together: $(x+1)(x-1) = x^2 - x - x + 1 = x^2 - 2x + 1$. This is the common denominator for the two fractions.

Simplifying the Expression

Now that we have the common denominator, we can rewrite each fraction with the common denominator:

$\frac{3}{x+1} = \frac{3(x-1)}{(x+1)(x-1)}$

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

Combining the Fractions

Now that we have both fractions with the common denominator, we can combine them by adding the numerators:

$\frac{3(x-1)}{(x+1)(x-1)} + \frac{x(x+1)}{(x+1)(x-1)} = \frac{3(x-1) + x(x+1)}{(x+1)(x-1)}$

Simplifying the Numerator

To simplify the numerator, we can expand the terms:

$3(x-1) + x(x+1) = 3x - 3 + x^2 + x$

Combining Like Terms

Now that we have the expanded numerator, we can combine like terms:

$3x - 3 + x^2 + x = x^2 + 4x - 3$

Simplifying the Expression

Now that we have the simplified numerator, we can rewrite the expression:

$\frac{x^2 + 4x - 3}{(x+1)(x-1)}$

Factoring the Numerator

To simplify the expression further, we can factor the numerator:

$x^2 + 4x - 3 = (x+1)(x-3)$

Simplifying the Expression

Now that we have factored the numerator, we can rewrite the expression:

$\frac{(x+1)(x-3)}{(x+1)(x-1)}$

Canceling Common Factors

Now that we have the expression in factored form, we can cancel common factors:

$\frac{(x+1)(x-3)}{(x+1)(x-1)} = \frac{x-3}{x-1}$

Conclusion

In this article, we simplified the expression $\frac{3}{x+1} + \frac{x}{x-1}$ using various techniques. We found the common denominator, combined the fractions, simplified the numerator, factored the numerator, and canceled common factors. The final simplified expression is $\frac{x-3}{x-1}$. This expression can be used to solve problems involving fractions and algebraic expressions.

Final Answer

The final answer is $\boxed{\frac{x-3}{x-1}}$.

Related Topics

• Simplifying expressions
• Combining fractions
• Factoring numerators
• Canceling common factors

Further Reading

• Simplifying expressions with variables
• Combining fractions with different denominators
• Factoring quadratic expressions
• Canceling common factors in algebraic expressions
  

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Introduction

In our previous article, we simplified the expression $\frac{3}{x+1} + \frac{x}{x-1}$ using various techniques. In this article, we will answer some frequently asked questions related to simplifying expressions and provide additional examples to help you understand the concept better.

Q&A

Q: What is the common denominator of two fractions?

A: The common denominator of two fractions is the product of the two denominators.

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

A: To find the common denominator, you can multiply the two denominators together.

Q: What is the difference between combining fractions and adding fractions?

A: Combining fractions involves finding a common denominator and adding the numerators, while adding fractions involves adding the numerators and keeping the same denominator.

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

A: Yes, you can simplify an expression with a variable in the denominator by finding a common denominator and combining the fractions.

Q: How do I factor a quadratic expression?

A: To factor a quadratic expression, you can look for two numbers whose product is the constant term and whose sum is the coefficient of the linear term.

Q: Can I cancel common factors in an expression with a variable in the denominator?

A: Yes, you can cancel common factors in an expression with a variable in the denominator, but you must be careful not to cancel a factor that is also in the denominator.

Q: What is the final simplified expression of $\frac{3}{x+1} + \frac{x}{x-1}$?

A: The final simplified expression of $\frac{3}{x+1} + \frac{x}{x-1}$ is $\frac{x-3}{x-1}$.

Additional Examples

Example 1: Simplify the expression $\frac{2}{x-2} + \frac{3}{x+2}$

To simplify this expression, we can find the common denominator, which is $(x-2)(x+2)$. Then, we can combine the fractions and simplify the numerator.

$\frac{2}{x-2} + \frac{3}{x+2} = \frac{2(x+2) + 3(x-2)}{(x-2)(x+2)}$

$= \frac{2x + 4 + 3x - 6}{(x-2)(x+2)}$

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

Example 2: Simplify the expression $\frac{4}{x+3} + \frac{2}{x-3}$

To simplify this expression, we can find the common denominator, which is $(x+3)(x-3)$. Then, we can combine the fractions and simplify the numerator.

$\frac{4}{x+3} + \frac{2}{x-3} = \frac{4(x-3) + 2(x+3)}{(x+3)(x-3)}$

$= \frac{4x - 12 + 2x + 6}{(x+3)(x-3)}$

$= \frac{6x - 6}{(x+3)(x-3)}$

Conclusion

In this article, we answered some frequently asked questions related to simplifying expressions and provided additional examples to help you understand the concept better. We also simplified two more expressions using the techniques we learned in our previous article. Remember to always find the common denominator, combine the fractions, and simplify the numerator to simplify an expression.

Final Answer

The final answer is $\boxed{\frac{x-3}{x-1}}$.

Related Topics

• Simplifying expressions
• Combining fractions
• Factoring numerators
• Canceling common factors

Further Reading

• Simplifying expressions with variables
• Combining fractions with different denominators
• Factoring quadratic expressions
• Canceling common factors in algebraic expressions