Which Expression Is Equivalent To The Following Complex Fraction?$\[ \frac{\frac{3}{x-1}-4}{2-\frac{2}{x-1}} \\]

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Understanding Complex Fractions

Complex fractions are mathematical expressions that contain fractions within fractions. They can be challenging to simplify, but with a clear understanding of the concept and a step-by-step approach, you can master the art of simplifying complex fractions. In this article, we will explore the process of simplifying complex fractions, using the given expression as a case study.

The Given Expression

The given expression is ${ \frac{\frac{3}{x-1}-4}{2-\frac{2}{x-1}} }$. This expression contains two fractions within fractions, making it a complex fraction. Our goal is to simplify this expression and find an equivalent expression that is easier to work with.

Step 1: Simplify the Numerator

To simplify the given expression, we start by simplifying the numerator. The numerator is ${ \frac{3}{x-1}-4 }$. We can simplify this expression by finding a common denominator for the two terms. The common denominator is x−1x-1, so we can rewrite the expression as ${ \frac{3}{x-1}-\frac{4(x-1)}{x-1} }$. Simplifying further, we get ${ \frac{3-4(x-1)}{x-1} }$.

Step 2: Simplify the Denominator

Next, we simplify the denominator of the given expression. The denominator is ${ 2-\frac{2}{x-1} }$. We can simplify this expression by finding a common denominator for the two terms. The common denominator is x−1x-1, so we can rewrite the expression as ${ \frac{2(x-1)-2}{x-1} }$. Simplifying further, we get ${ \frac{2x-2-2}{x-1} }$.

Step 3: Combine the Numerator and Denominator

Now that we have simplified the numerator and denominator, we can combine them to get the simplified expression. The simplified expression is ${ \frac{\frac{3-4(x-1)}{x-1}}{\frac{2x-4}{x-1}} }$.

Step 4: Simplify the Expression

To simplify the expression further, we can cancel out any common factors in the numerator and denominator. In this case, we can cancel out the factor of x−1x-1 in the numerator and denominator. This gives us the simplified expression ${ \frac{3-4x+4}{2x-4} }$.

Step 5: Simplify the Expression Further

We can simplify the expression further by combining like terms in the numerator. The numerator is 3−4x+43-4x+4, which simplifies to 7−4x7-4x. The simplified expression is ${ \frac{7-4x}{2x-4} }$.

Step 6: Factor the Expression

To simplify the expression further, we can factor the numerator and denominator. The numerator is 7−4x7-4x, which can be factored as −4(x−7/4)-4(x-7/4). The denominator is 2x−42x-4, which can be factored as 2(x−2)2(x-2). The simplified expression is ${ \frac{-4(x-7/4)}{2(x-2)} }$.

Step 7: Cancel Out Common Factors

Finally, we can cancel out any common factors in the numerator and denominator. In this case, we can cancel out the factor of −2-2 in the numerator and denominator. This gives us the simplified expression ${ \frac{2(x-7/4)}{(x-2)} }$.

Conclusion

In this article, we have simplified the given complex fraction expression using a step-by-step approach. We started by simplifying the numerator and denominator, then combined them to get the simplified expression. We then simplified the expression further by canceling out common factors and factoring the numerator and denominator. The final simplified expression is ${ \frac{2(x-7/4)}{(x-2)} }$. This expression is equivalent to the original complex fraction expression, but is much easier to work with.

Real-World Applications

Simplifying complex fractions is an important skill in mathematics, with many real-world applications. For example, in physics, complex fractions are used to describe the motion of objects in terms of their position, velocity, and acceleration. In engineering, complex fractions are used to design and analyze complex systems, such as electrical circuits and mechanical systems. In finance, complex fractions are used to calculate interest rates and investment returns.

Tips and Tricks

Simplifying complex fractions can be challenging, but with practice and patience, you can master the art. Here are some tips and tricks to help you simplify complex fractions:

  • Start by simplifying the numerator and denominator separately.
  • Use a common denominator to combine the numerator and denominator.
  • Cancel out any common factors in the numerator and denominator.
  • Factor the numerator and denominator to simplify the expression further.
  • Use algebraic manipulations to simplify the expression.

Common Mistakes

When simplifying complex fractions, it's easy to make mistakes. Here are some common mistakes to avoid:

  • Failing to simplify the numerator and denominator separately.
  • Not using a common denominator to combine the numerator and denominator.
  • Not canceling out common factors in the numerator and denominator.
  • Not factoring the numerator and denominator to simplify the expression further.
  • Not using algebraic manipulations to simplify the expression.

Conclusion

Simplifying complex fractions is an important skill in mathematics, with many real-world applications. By following a step-by-step approach and using algebraic manipulations, you can simplify complex fractions and find equivalent expressions that are easier to work with. With practice and patience, you can master the art of simplifying complex fractions and apply it to a wide range of mathematical and real-world problems.