Solve The Equation:$\[ \frac{x}{x-1}=\frac{x-2}{x+1} \\]What Is The Solution Set Of The Equation Above?A) -2 B) \[$-\frac{1}{2}\$\] C) \[$\frac{1}{2}\$\] D) 2

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Introduction

In this article, we will delve into the world of algebra and solve a seemingly complex equation. The equation in question is xx−1=x−2x+1\frac{x}{x-1}=\frac{x-2}{x+1}. Our goal is to find the solution set of this equation, which means identifying all the values of xx that satisfy the equation.

Understanding the Equation

Before we dive into the solution, let's take a closer look at the equation. We have two fractions on both sides of the equation, and our task is to find the values of xx that make both sides equal.

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

Step 1: Cross-Multiplication

One way to solve this equation is to use cross-multiplication. This involves multiplying both sides of the equation by the denominators of both fractions, which are (x−1)(x-1) and (x+1)(x+1).

(x)(x+1)=(x−2)(x−1)(x)(x+1) = (x-2)(x-1)

Step 2: Expanding the Equation

Now that we have cross-multiplied, let's expand both sides of the equation.

x2+x=x2−3x+2x^2 + x = x^2 - 3x + 2

Step 3: Simplifying the Equation

We can simplify the equation by combining like terms.

x2+x−x2+3x−2=0x^2 + x - x^2 + 3x - 2 = 0

This simplifies to:

4x−2=04x - 2 = 0

Step 4: Solving for x

Now that we have a simplified equation, let's solve for xx.

4x−2=04x - 2 = 0

Adding 2 to both sides gives us:

4x=24x = 2

Dividing both sides by 4 gives us:

x=12x = \frac{1}{2}

Conclusion

And there you have it! The solution set of the equation xx−1=x−2x+1\frac{x}{x-1}=\frac{x-2}{x+1} is x=12x = \frac{1}{2}. This means that the only value of xx that satisfies the equation is 12\frac{1}{2}.

Final Answer

The final answer is 12\boxed{\frac{1}{2}}.

Discussion

Now that we have solved the equation, let's take a step back and analyze the solution. We used cross-multiplication and simplification to arrive at the solution x=12x = \frac{1}{2}. This solution makes sense, as it satisfies the original equation.

Alternative Solutions

It's worth noting that there may be alternative solutions to this equation. However, in this case, we have found a single solution that satisfies the equation.

Conclusion

In conclusion, solving the equation xx−1=x−2x+1\frac{x}{x-1}=\frac{x-2}{x+1} requires careful manipulation of the equation using cross-multiplication and simplification. The solution set of the equation is x=12x = \frac{1}{2}.

Final Thoughts

Solving equations like this one requires patience and attention to detail. However, with practice and experience, you can develop the skills necessary to solve even the most complex equations.

Common Mistakes

When solving equations like this one, it's easy to make mistakes. Some common mistakes include:

  • Not simplifying the equation enough
  • Not checking for extraneous solutions
  • Not using the correct method to solve the equation

Tips and Tricks

Here are some tips and tricks to help you solve equations like this one:

  • Always start by simplifying the equation
  • Use cross-multiplication to eliminate the fractions
  • Check for extraneous solutions
  • Use the correct method to solve the equation

Conclusion

In conclusion, solving the equation xx−1=x−2x+1\frac{x}{x-1}=\frac{x-2}{x+1} requires careful manipulation of the equation using cross-multiplication and simplification. The solution set of the equation is x=12x = \frac{1}{2}. With practice and experience, you can develop the skills necessary to solve even the most complex equations.

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Introduction

In our previous article, we solved the equation xx−1=x−2x+1\frac{x}{x-1}=\frac{x-2}{x+1} and found the solution set to be x=12x = \frac{1}{2}. However, we know that there are often many questions and doubts that arise when solving equations. In this article, we will address some of the most frequently asked questions about solving the equation xx−1=x−2x+1\frac{x}{x-1}=\frac{x-2}{x+1}.

Q&A

Q: What is the first step in solving the equation xx−1=x−2x+1\frac{x}{x-1}=\frac{x-2}{x+1}?

A: The first step in solving the equation is to cross-multiply. This involves multiplying both sides of the equation by the denominators of both fractions, which are (x−1)(x-1) and (x+1)(x+1).

Q: Why do we need to cross-multiply?

A: We need to cross-multiply to eliminate the fractions and make it easier to solve the equation.

Q: What is the next step after cross-multiplying?

A: After cross-multiplying, we need to expand both sides of the equation and simplify it.

Q: How do we simplify the equation?

A: We simplify the equation by combining like terms and eliminating any unnecessary terms.

Q: What is the final step in solving the equation?

A: The final step in solving the equation is to solve for xx.

Q: How do we solve for xx?

A: We solve for xx by isolating the variable xx on one side of the equation.

Q: What is the solution set of the equation xx−1=x−2x+1\frac{x}{x-1}=\frac{x-2}{x+1}?

A: The solution set of the equation is x=12x = \frac{1}{2}.

Q: Why is it important to check for extraneous solutions?

A: It is important to check for extraneous solutions because some solutions may not satisfy the original equation.

Q: How do we check for extraneous solutions?

A: We check for extraneous solutions by plugging the solution back into the original equation and checking if it is true.

Q: What are some common mistakes to avoid when solving equations?

A: Some common mistakes to avoid when solving equations include not simplifying the equation enough, not checking for extraneous solutions, and not using the correct method to solve the equation.

Q: What are some tips and tricks for solving equations?

A: Some tips and tricks for solving equations include always starting by simplifying the equation, using cross-multiplication to eliminate the fractions, checking for extraneous solutions, and using the correct method to solve the equation.

Conclusion

In conclusion, solving the equation xx−1=x−2x+1\frac{x}{x-1}=\frac{x-2}{x+1} requires careful manipulation of the equation using cross-multiplication and simplification. By following the steps outlined in this article, you can develop the skills necessary to solve even the most complex equations.

Final Thoughts

Solving equations like this one requires patience and attention to detail. However, with practice and experience, you can develop the skills necessary to solve even the most complex equations.

Common Mistakes

When solving equations like this one, it's easy to make mistakes. Some common mistakes include:

  • Not simplifying the equation enough
  • Not checking for extraneous solutions
  • Not using the correct method to solve the equation

Tips and Tricks

Here are some tips and tricks to help you solve equations like this one:

  • Always start by simplifying the equation
  • Use cross-multiplication to eliminate the fractions
  • Check for extraneous solutions
  • Use the correct method to solve the equation

Conclusion

In conclusion, solving the equation xx−1=x−2x+1\frac{x}{x-1}=\frac{x-2}{x+1} requires careful manipulation of the equation using cross-multiplication and simplification. By following the steps outlined in this article, you can develop the skills necessary to solve even the most complex equations.