Simplify The Expression: 1 1 − X + 2 1 + X \frac{1}{1-x} + \frac{2}{1+x} 1 − X 1 + 1 + X 2

Mar 11, 2025 by ADMIN 95 views

$Simplify the Expression: $\frac{1}{1-x} + \frac{2}{1+x}$$

Introduction

In mathematics, simplifying expressions is a crucial step in solving problems and understanding complex concepts. The given expression, $\frac{1}{1-x} + \frac{2}{1+x}$ , is a combination of two fractions with different denominators. To simplify this expression, we need to find a common denominator and combine the fractions. In this article, we will walk you through the step-by-step process of simplifying the given expression.

Understanding the Expression

The given expression consists of two fractions: $\frac{1}{1-x}$ and $\frac{2}{1+x}$ . The first fraction has a denominator of $1-x$ , while the second fraction has a denominator of $1+x$ . To simplify the expression, we need to find a common denominator for both fractions.

Finding a Common Denominator

To find a common denominator, we need to identify the least common multiple (LCM) of the two denominators. In this case, the LCM of $1-x$ and $1+x$ is $(1-x)(1+x)$ . This is because the product of two binomials, $(1-x)$ and $(1+x)$ , results in a quadratic expression, $1-x^2$ , which is the LCM of the two denominators.

Simplifying the Expression

Now that we have found the common denominator, we can rewrite the expression with the common denominator. We will multiply the numerator and denominator of the first fraction by $(1+x)$ , and the numerator and denominator of the second fraction by $(1-x)$ .

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

Combining the Fractions

Now that we have rewritten the expression with the common denominator, we can combine the fractions by adding the numerators.

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

Simplifying the Numerator

We can simplify the numerator by combining like terms.

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

Simplifying the Expression Further

We can simplify the numerator further by combining like terms.

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

Final Simplification

We can simplify the expression further by factoring the numerator.

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

Conclusion

In this article, we simplified the expression $\frac{1}{1-x} + \frac{2}{1+x}$ by finding a common denominator and combining the fractions. We then simplified the numerator by combining like terms and factored the numerator to obtain the final simplified expression. This process demonstrates the importance of simplifying expressions in mathematics and provides a step-by-step guide for simplifying complex expressions.

Common Applications

Simplifying expressions is a crucial step in solving problems in various fields, including:

Algebra: Simplifying expressions is essential in solving linear and quadratic equations.
Calculus: Simplifying expressions is necessary in finding derivatives and integrals.
Physics: Simplifying expressions is crucial in solving problems involving motion, energy, and momentum.
Engineering: Simplifying expressions is necessary in designing and analyzing complex systems.

Tips and Tricks

When simplifying expressions, it is essential to:

Identify the common denominator and rewrite the expression with the common denominator.
Combine the fractions by adding the numerators.
Simplify the numerator by combining like terms.
Factor the numerator to obtain the final simplified expression.

By following these tips and tricks, you can simplify complex expressions and solve problems in various fields.

Real-World Examples

Simplifying expressions is essential in solving real-world problems. For example:

In physics, simplifying expressions is necessary in solving problems involving motion, energy, and momentum. For instance, the equation of motion for an object under constant acceleration is given by $s = ut + \frac{1}{2}at^2$ , where $s$ is the displacement, $u$ is the initial velocity, $t$ is the time, and $a$ is the acceleration. Simplifying this expression is essential in solving problems involving motion.
In engineering, simplifying expressions is necessary in designing and analyzing complex systems. For instance, the equation for the stress on a beam is given by $\sigma = \frac{F}{A}$ , where $\sigma$ is the stress, $F$ is the force, and $A$ is the cross-sectional area. Simplifying this expression is essential in designing and analyzing complex systems.

Final Thoughts

Simplifying expressions is a crucial step in solving problems in various fields. By following the step-by-step process outlined in this article, you can simplify complex expressions and solve problems in algebra, calculus, physics, and engineering. Remember to identify the common denominator, combine the fractions, simplify the numerator, and factor the numerator to obtain the final simplified expression. With practice and patience, you can become proficient in simplifying expressions and solving complex problems.

Introduction

In our previous article, we simplified the expression $\frac{1}{1-x} + \frac{2}{1+x}$ by finding a common denominator and combining the fractions. We then simplified the numerator by combining like terms and factored the numerator to obtain the final simplified expression. In this article, we will answer some frequently asked questions (FAQs) related to simplifying expressions.

Q&A

Q: What is the common denominator of two fractions?

A: The common denominator of two fractions is the least common multiple (LCM) of the two denominators. In the case of the expression $\frac{1}{1-x} + \frac{2}{1+x}$ , the common denominator is $(1-x)(1+x)$ .

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

A: To find the common denominator of two fractions, you need to identify the least common multiple (LCM) of the two denominators. You can do this by listing the multiples of each denominator and finding the smallest multiple that is common to both.

Q: What is the difference between a common denominator and a least common multiple (LCM)?

A: A common denominator is the denominator that is common to two or more fractions, while a least common multiple (LCM) is the smallest multiple that is common to two or more numbers. In the case of the expression $\frac{1}{1-x} + \frac{2}{1+x}$ , the common denominator is $(1-x)(1+x)$ , while the LCM of the two denominators is also $(1-x)(1+x)$ .

Q: How do I simplify a fraction with a variable in the denominator?

A: To simplify a fraction with a variable in the denominator, you need to find a common denominator and combine the fractions. You can then simplify the numerator by combining like terms and factoring the numerator to obtain the final simplified expression.

Q: What is the difference between simplifying an expression and solving an equation?

A: Simplifying an expression involves combining like terms and factoring the numerator to obtain the final simplified expression, while solving an equation involves finding the value of the variable that makes the equation true.

Q: How do I know when to simplify an expression?

A: You should simplify an expression when it is necessary to make the expression easier to work with or when it is necessary to solve a problem. Simplifying an expression can help you to identify patterns and relationships between variables, which can be useful in solving problems.

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

A: Yes, you can simplify an expression with a negative exponent by rewriting the expression with a positive exponent. For example, the expression $x^{-2}$ can be rewritten as $\frac{1}{x^2}$ .

Q: How do I simplify an expression with a fraction in the numerator?

A: To simplify an expression with a fraction in the numerator, you need to find a common denominator and combine the fractions. You can then simplify the numerator by combining like terms and factoring the numerator to obtain the final simplified expression.

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

A: Yes, you can simplify an expression with a variable in the numerator by combining like terms and factoring the numerator to obtain the final simplified expression.

Conclusion

Simplifying expressions is an essential step in solving problems in various fields. By following the step-by-step process outlined in this article, you can simplify complex expressions and solve problems in algebra, calculus, physics, and engineering. Remember to identify the common denominator, combine the fractions, simplify the numerator, and factor the numerator to obtain the final simplified expression. With practice and patience, you can become proficient in simplifying expressions and solving complex problems.

Common Applications

Simplifying expressions is a crucial step in solving problems in various fields, including:

Algebra: Simplifying expressions is essential in solving linear and quadratic equations.
Calculus: Simplifying expressions is necessary in finding derivatives and integrals.
Physics: Simplifying expressions is crucial in solving problems involving motion, energy, and momentum.
Engineering: Simplifying expressions is necessary in designing and analyzing complex systems.

Tips and Tricks

When simplifying expressions, it is essential to:

Identify the common denominator and rewrite the expression with the common denominator.
Combine the fractions by adding the numerators.
Simplify the numerator by combining like terms.
Factor the numerator to obtain the final simplified expression.

By following these tips and tricks, you can simplify complex expressions and solve problems in various fields.

Real-World Examples

Simplifying expressions is essential in solving real-world problems. For example:

In physics, simplifying expressions is necessary in solving problems involving motion, energy, and momentum. For instance, the equation of motion for an object under constant acceleration is given by $s = ut + \frac{1}{2}at^2$ , where $s$ is the displacement, $u$ is the initial velocity, $t$ is the time, and $a$ is the acceleration. Simplifying this expression is essential in solving problems involving motion.
In engineering, simplifying expressions is necessary in designing and analyzing complex systems. For instance, the equation for the stress on a beam is given by $\sigma = \frac{F}{A}$ , where $\sigma$ is the stress, $F$ is the force, and $A$ is the cross-sectional area. Simplifying this expression is essential in designing and analyzing complex systems.