Which Expression Is Equivalent To The Following Complex Fraction?$\[ \frac{\frac{3}{x-1}-4}{2-\frac{2}{x-1}} \\]A. \[$\frac{2(x-2)}{-4x+7}\$\]B. \[$\frac{-4x+7}{2(x-2)}\$\]C. \[$\frac{-4x+7}{2\left(x^2-2\right)}\$\]D.

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Understanding Complex Fractions

A complex fraction is a fraction that contains one or more fractions in its numerator or denominator. These types of fractions can be challenging to simplify, but with the right approach, they can be broken down into simpler expressions. In this article, we will explore how to simplify a complex fraction and apply this knowledge to a specific problem.

The Problem: Simplifying a Complex Fraction

The problem we will be working with is the following complex fraction:

{ \frac{\frac{3}{x-1}-4}{2-\frac{2}{x-1}} \}

Our goal is to simplify this complex fraction and determine which of the given expressions is equivalent to it.

Step 1: Simplify the Numerator

To simplify the complex fraction, we start by simplifying the numerator. The numerator is a fraction with a fraction in its numerator and a constant in its denominator. We can simplify this by finding a common denominator for the two fractions.

{ \frac{3}{x-1}-4 = \frac{3}{x-1}-\frac{4(x-1)}{x-1} \}

Combining the two fractions, we get:

{ \frac{3-4(x-1)}{x-1} \}

Expanding the numerator, we get:

{ \frac{3-4x+4}{x-1} \}

Simplifying the numerator further, we get:

{ \frac{-4x+7}{x-1} \}

Step 2: Simplify the Denominator

Next, we simplify the denominator. The denominator is a fraction with a constant in its numerator and a fraction in its denominator. We can simplify this by finding a common denominator for the two fractions.

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

Combining the two fractions, we get:

{ \frac{2(x-1)-2}{x-1} \}

Simplifying the numerator, we get:

{ \frac{2x-2-2}{x-1} \}

Simplifying further, we get:

{ \frac{2x-4}{x-1} \}

Step 3: Simplify the Complex Fraction

Now that we have simplified the numerator and denominator, we can simplify the complex fraction by dividing the numerator by the denominator.

{ \frac{\frac{-4x+7}{x-1}}{\frac{2x-4}{x-1}} \}

To divide fractions, we multiply the numerator by the reciprocal of the denominator.

{ \frac{-4x+7}{x-1} \cdot \frac{x-1}{2x-4} \}

Simplifying the expression, we get:

{ \frac{-4x+7}{2x-4} \}

Step 4: Factor the Denominator

The denominator can be factored as follows:

{ 2x-4 = 2(x-2) \}

Step 5: Simplify the Expression

Now that we have factored the denominator, we can simplify the expression by canceling out any common factors.

{ \frac{-4x+7}{2(x-2)} \}

Conclusion

In conclusion, the complex fraction ${ \frac{\frac{3}{x-1}-4}{2-\frac{2}{x-1}} }$ is equivalent to the expression ${ \frac{-4x+7}{2(x-2)} }$. This expression can be further simplified by factoring the denominator, but it is already in its simplest form.

Answer

The correct answer is:

{ \boxed{\frac{-4x+7}{2(x-2)}} \}

This expression is equivalent to the complex fraction ${ \frac{\frac{3}{x-1}-4}{2-\frac{2}{x-1}} }$.

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Q: What is a complex fraction?

A complex fraction is a fraction that contains one or more fractions in its numerator or denominator. These types of fractions can be challenging to simplify, but with the right approach, they can be broken down into simpler expressions.

Q: How do I simplify a complex fraction?

To simplify a complex fraction, you need to follow these steps:

  1. Simplify the numerator by finding a common denominator for the two fractions.
  2. Simplify the denominator by finding a common denominator for the two fractions.
  3. Divide the numerator by the denominator by multiplying the numerator by the reciprocal of the denominator.
  4. Simplify the expression by canceling out any common factors.

Q: What is the difference between a complex fraction and a simple fraction?

A simple fraction is a fraction that contains only numbers in its numerator and denominator. A complex fraction, on the other hand, contains one or more fractions in its numerator or denominator.

Q: Can I simplify a complex fraction by canceling out common factors?

Yes, you can simplify a complex fraction 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: How do I know if a complex fraction can be simplified?

A complex fraction can be simplified if it meets the following conditions:

  • The numerator and denominator have a common factor.
  • The numerator and denominator can be factored into simpler expressions.
  • The expression can be simplified by canceling out common factors.

Q: What are some common mistakes to avoid when simplifying complex fractions?

Some common mistakes to avoid when simplifying complex fractions include:

  • Canceling out factors that are not common to both the numerator and denominator.
  • Not simplifying the numerator and denominator separately.
  • Not using the correct order of operations when simplifying the expression.

Q: Can I use a calculator to simplify complex fractions?

Yes, you can use a calculator to simplify complex fractions. However, it's always a good idea to check your work by simplifying the expression manually to ensure that you get the correct answer.

Q: How do I check my work when simplifying complex fractions?

To check your work when simplifying complex fractions, you can:

  • Simplify the expression manually to ensure that you get the correct answer.
  • Use a calculator to check your work.
  • Compare your answer to the original expression to ensure that it is equivalent.

Q: What are some real-world applications of simplifying complex fractions?

Simplifying complex fractions has many real-world applications, including:

  • Calculating probabilities and statistics.
  • Solving systems of equations.
  • Modeling real-world phenomena, such as population growth and chemical reactions.

Q: Can I use simplifying complex fractions to solve word problems?

Yes, you can use simplifying complex fractions to solve word problems. By simplifying complex fractions, you can break down complex problems into simpler expressions that are easier to work with.

Q: How do I apply simplifying complex fractions to solve word problems?

To apply simplifying complex fractions to solve word problems, you can:

  • Read the problem carefully and identify the key elements.
  • Simplify the complex fractions in the problem.
  • Use the simplified expressions to solve the problem.

Q: What are some common word problems that involve simplifying complex fractions?

Some common word problems that involve simplifying complex fractions include:

  • Calculating probabilities and statistics.
  • Solving systems of equations.
  • Modeling real-world phenomena, such as population growth and chemical reactions.

Q: Can I use simplifying complex fractions to solve algebraic equations?

Yes, you can use simplifying complex fractions to solve algebraic equations. By simplifying complex fractions, you can break down complex equations into simpler expressions that are easier to work with.

Q: How do I apply simplifying complex fractions to solve algebraic equations?

To apply simplifying complex fractions to solve algebraic equations, you can:

  • Simplify the complex fractions in the equation.
  • Use the simplified expressions to solve the equation.

Q: What are some common algebraic equations that involve simplifying complex fractions?

Some common algebraic equations that involve simplifying complex fractions include:

  • Linear equations.
  • Quadratic equations.
  • Polynomial equations.

Q: Can I use simplifying complex fractions to solve trigonometric equations?

Yes, you can use simplifying complex fractions to solve trigonometric equations. By simplifying complex fractions, you can break down complex equations into simpler expressions that are easier to work with.

Q: How do I apply simplifying complex fractions to solve trigonometric equations?

To apply simplifying complex fractions to solve trigonometric equations, you can:

  • Simplify the complex fractions in the equation.
  • Use the simplified expressions to solve the equation.

Q: What are some common trigonometric equations that involve simplifying complex fractions?

Some common trigonometric equations that involve simplifying complex fractions include:

  • Trigonometric identities.
  • Trigonometric equations involving sine, cosine, and tangent.

Q: Can I use simplifying complex fractions to solve exponential equations?

Yes, you can use simplifying complex fractions to solve exponential equations. By simplifying complex fractions, you can break down complex equations into simpler expressions that are easier to work with.

Q: How do I apply simplifying complex fractions to solve exponential equations?

To apply simplifying complex fractions to solve exponential equations, you can:

  • Simplify the complex fractions in the equation.
  • Use the simplified expressions to solve the equation.

Q: What are some common exponential equations that involve simplifying complex fractions?

Some common exponential equations that involve simplifying complex fractions include:

  • Exponential growth and decay equations.
  • Exponential equations involving logarithms.

Q: Can I use simplifying complex fractions to solve logarithmic equations?

Yes, you can use simplifying complex fractions to solve logarithmic equations. By simplifying complex fractions, you can break down complex equations into simpler expressions that are easier to work with.

Q: How do I apply simplifying complex fractions to solve logarithmic equations?

To apply simplifying complex fractions to solve logarithmic equations, you can:

  • Simplify the complex fractions in the equation.
  • Use the simplified expressions to solve the equation.

Q: What are some common logarithmic equations that involve simplifying complex fractions?

Some common logarithmic equations that involve simplifying complex fractions include:

  • Logarithmic growth and decay equations.
  • Logarithmic equations involving exponential functions.

Q: Can I use simplifying complex fractions to solve systems of equations?

Yes, you can use simplifying complex fractions to solve systems of equations. By simplifying complex fractions, you can break down complex equations into simpler expressions that are easier to work with.

Q: How do I apply simplifying complex fractions to solve systems of equations?

To apply simplifying complex fractions to solve systems of equations, you can:

  • Simplify the complex fractions in the equations.
  • Use the simplified expressions to solve the system of equations.

Q: What are some common systems of equations that involve simplifying complex fractions?

Some common systems of equations that involve simplifying complex fractions include:

  • Linear systems of equations.
  • Quadratic systems of equations.
  • Polynomial systems of equations.

Q: Can I use simplifying complex fractions to solve differential equations?

Yes, you can use simplifying complex fractions to solve differential equations. By simplifying complex fractions, you can break down complex equations into simpler expressions that are easier to work with.

Q: How do I apply simplifying complex fractions to solve differential equations?

To apply simplifying complex fractions to solve differential equations, you can:

  • Simplify the complex fractions in the equation.
  • Use the simplified expressions to solve the differential equation.

Q: What are some common differential equations that involve simplifying complex fractions?

Some common differential equations that involve simplifying complex fractions include:

  • Ordinary differential equations (ODEs).
  • Partial differential equations (PDEs).

Q: Can I use simplifying complex fractions to solve partial differential equations?

Yes, you can use simplifying complex fractions to solve partial differential equations. By simplifying complex fractions, you can break down complex equations into simpler expressions that are easier to work with.

Q: How do I apply simplifying complex fractions to solve partial differential equations?

To apply simplifying complex fractions to solve partial differential equations, you can:

  • Simplify the complex fractions in the equation.
  • Use the simplified expressions to solve the partial differential equation.

Q: What are some common partial differential equations that involve simplifying complex fractions?

Some common partial differential equations that involve simplifying complex fractions include:

  • Heat equation.
  • Wave equation.
  • Laplace equation.

Q: Can I use simplifying complex fractions to solve stochastic differential equations?

Yes, you can use simplifying complex fractions to solve stochastic differential equations. By simplifying complex fractions, you can break down complex equations into simpler expressions that are easier to work with.

Q: How do I apply simplifying complex fractions to solve stochastic differential equations?

To apply simplifying complex fractions to solve stochastic differential equations, you can:

  • Simplify the complex fractions in the equation.
  • Use the simplified expressions to solve the stochastic differential equation.

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