Dividing Rational Expressions: Step-by-Step Solution

Hey guys! Let's dive into a common math problem: dividing rational expressions. Specifically, we're going to break down the expression ${\frac{30}{3x-5} ÷ \frac{5x}{x^2-4}.}$ This might look a little intimidating at first, but don't worry, we'll take it step by step. Understanding how to divide rational expressions is crucial not just for algebra, but also for more advanced math courses like calculus. It's a fundamental skill that helps simplify complex equations and solve real-world problems. In this article, we'll not only solve the problem but also understand the underlying concepts and techniques. This includes how to factor polynomials, identify restrictions on variables, and simplify the final result. So, grab your pencils, and let's get started!

Understanding Rational Expressions

Before we tackle the division, let's make sure we're all on the same page about what rational expressions are. Rational expressions are essentially fractions where the numerator and the denominator are polynomials. Think of them as the algebraic cousins of regular numerical fractions. For instance, ${\frac{30}{3x-5}}$ and ${\frac{5x}{x^2-4}}$ are both rational expressions. The key thing to remember with rational expressions is that the denominator cannot be zero. This is because division by zero is undefined in mathematics. This leads us to an important concept: restrictions on the variable.

Identifying Restrictions

To identify restrictions, we need to find the values of ${x}$ that would make the denominator zero. For the expression ${\frac{30}{3x-5}}$, we set the denominator equal to zero and solve for ${x}$:

${3x - 5 = 0}$

${3x = 5}$

${x = \frac{5}{3}}$

So, ${x}$ cannot be ${\frac{5}{3}}$. For the second expression, ${\frac{5x}{x^2-4}}$, we have:

${x^2 - 4 = 0}$

This is a difference of squares, which factors to:

${(x - 2)(x + 2) = 0}$

Thus, ${x}$ cannot be 2 or -2. These restrictions are super important because they define the domain of the expression. Ignoring them can lead to incorrect solutions or misinterpretations of the problem. In our original problem, these restrictions will come into play when we simplify and solve. By understanding these restrictions, we ensure our solution is mathematically sound and applicable. These foundational concepts are essential for mastering more complex algebraic manipulations.

The Division Process: Keep, Change, Flip

Now, let's get to the heart of the problem: dividing rational expressions. The golden rule for dividing fractions, whether they're numerical or algebraic, is