Simplify This Complex Fraction: A Math Problem

Dealing with complex fractions can sometimes feel like navigating a maze, but with a clear strategy, simplifying them becomes a straightforward process. The complex fraction we’re looking at today is:

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

Our goal is to find which of the given expressions is equivalent to this one. Let's break it down step by step, making sure to keep our mathematical footing firm. Remember, the key to simplifying fractions, especially complex ones, is to combine terms in the numerator and the denominator separately before tackling the division.

Understanding Complex Fractions

A complex fraction is essentially a fraction where the numerator, the denominator, or both, contain fractions themselves. In our case, both the numerator (rac{3}{x-1}-4) and the denominator (2-rac{2}{x-1}) are expressions that involve fractions. To simplify this, we need to transform these into single, simple fractions. This is often achieved by finding a common denominator for the terms within the numerator and the denominator.

Let's focus on the numerator first: $\frac{3}{x-1}-4$. To combine these, we need a common denominator, which is clearly $(x-1)$. We can rewrite $4$ as $\frac{4(x-1)}{x-1}$. So, the numerator becomes:

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

Now, let's distribute the $-4$ in the numerator: $3 - 4x + 4$. This simplifies to $7 - 4x$. So, the numerator is $\frac{7 - 4x}{x-1}$.

Next, let's tackle the denominator: $2-\frac{2}{x-1}$. Again, we need a common denominator, which is $(x-1)$. We rewrite $2$ as $\frac{2(x-1)}{x-1}$. The denominator then becomes:

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

Let's distribute the $2$ in the numerator of this fraction: $2x - 2 - 2$. This simplifies to $2x - 4$. So, the denominator is $\frac{2x - 4}{x-1}$.

Performing the Division

Now that we have simplified both the numerator and the denominator into single fractions, we can rewrite the original complex fraction as a division problem:

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

Dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of $\frac{2x - 4}{x-1}$ is $\frac{x-1}{2x - 4}$. So, our expression becomes:

$$\frac{7 - 4x}{x-1} \times \frac{x-1}{2x - 4}$$

Notice that the $(x-1)$ terms in the numerator and the denominator cancel each other out. This leaves us with:

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

Final Simplification and Matching Options

We are almost there! We can factor out a $2$ from the denominator $(2x - 4)$ to get $2(x-2)$. The expression is now:

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

Let's compare this to the given options. We see that the numerator $7 - 4x$ is the same as $-4x + 7$. So, the expression is $\frac{-4x + 7}{2(x - 2)}$.

Looking at the options provided:

A. $\frac{2(x-2)}{-4 x+7}$ B. $\frac{-4 x+7}{2(x-2)}$ C. $\frac{-4 x+7}{2\left(x^2-2\right)}$

Our simplified expression $\frac{-4x + 7}{2(x - 2)}$ exactly matches Option B. Therefore, Option B is the correct answer.

Why This Matters in Mathematics

Simplifying complex fractions is a fundamental skill in algebra. It not only helps in solving equations and inequalities but also forms the basis for more advanced mathematical concepts, such as calculus and linear algebra. Understanding how to manipulate these expressions demonstrates a strong grasp of basic arithmetic operations applied to algebraic terms. The ability to identify common denominators, perform fraction addition and subtraction, and understand the concept of reciprocals are all crucial components that are reinforced through such exercises.

Moreover, this process often involves careful attention to detail, especially when dealing with negative signs and variable expressions. A slight error in distributing a negative or in finding a common denominator can lead to an incorrect final answer. This reinforces the importance of methodical problem-solving and double-checking each step. The cancellation of terms, like the $(x-1)$ in our example, is a powerful visual cue that we are on the right track, showcasing the elegance of algebraic simplification. The goal is always to reduce a complicated expression to its simplest form, making it easier to analyze, understand, and use in further calculations.

Beyond the immediate task of finding an equivalent expression, mastering complex fractions builds confidence in tackling more intricate algebraic problems. It’s a stepping stone that prepares students for the challenges of higher mathematics, where the ability to simplify and manipulate algebraic expressions is paramount. The practice ensures that students are not just memorizing formulas but are developing a deep, intuitive understanding of how mathematical expressions work and can be transformed. This skill is invaluable not just in academic settings but in any field that relies on logical reasoning and quantitative analysis.

Conclusion

We successfully simplified the given complex fraction by systematically combining terms in the numerator and the denominator using common denominators, and then performing the division. This methodical approach led us to identify that Option B, $\frac{-4 x+7}{2(x-2)}$, is the expression equivalent to the original complex fraction. Keep practicing these types of problems, and you'll find your confidence and speed in algebraic manipulation will grow significantly!

For further exploration into algebraic simplification and complex fractions, you can visit Khan Academy's Algebra section, a fantastic resource for learning and reinforcing mathematical concepts.