Simplify (2(3x+2))/(9x^2-4) - 2/(3x+1): A Step-by-Step Guide

Hey guys! Today, we're diving into a fun little math problem where we'll simplify a complex fraction. Our mission is to express the expression $\frac{2(3x+2)}{9x^2-4} - \frac{2}{3x+1}$ as a single fraction in its simplest form. Trust me, it's not as daunting as it looks! We'll break it down step by step, so you can follow along easily. Get your pencils and paper ready, and let's get started!

Understanding the Problem

Before we jump into solving, let’s take a moment to really understand what we're dealing with. The expression we need to simplify is:

$\frac{2(3x+2)}{9x^2-4} - \frac{2}{3x+1}$

This looks like a subtraction of two fractions. To subtract fractions, we need a common denominator. Notice anything interesting about the denominators? The first denominator, 9x² - 4, looks like a difference of squares. Spotting these patterns is key in simplifying algebraic expressions. The second denominator is 3x + 1, which seems simpler but might be related to the first one. Our main goal here is to combine these two fractions into one single fraction and make it as simple as possible. This usually involves factoring, finding common denominators, combining terms, and canceling out any common factors. It’s like a puzzle, and we’re about to solve it!

Factoring the Denominator

The first step in simplifying this expression is to factor the denominator of the first fraction, which is 9x² - 4. As we noticed earlier, this is a difference of squares. Remember the difference of squares formula? It says that a² - b² = (a + b)(a - b). In our case, 9x² can be seen as (3x)², and 4 is 2². So, we can rewrite 9x² - 4 as (3x)² - 2². Applying the difference of squares formula, we get:

9x² - 4 = (3x + 2)(3x - 2)

Now, let's rewrite our original expression with the factored denominator:

$\frac{2(3x+2)}{(3x+2)(3x-2)} - \frac{2}{3x+1}$

See how (3x + 2) appears in both the numerator and the denominator of the first fraction? That’s a big clue! We can simplify the first fraction even further. Factoring is such a powerful tool in simplifying algebraic expressions. It allows us to see the structure more clearly and identify common factors that can be canceled out. By recognizing the difference of squares pattern, we’ve already made significant progress in simplifying our expression. Next, we'll cancel out those common factors and move on to finding a common denominator for both fractions.

Simplifying the First Fraction

Now that we’ve factored the denominator of the first fraction, we can simplify it. Our expression currently looks like this:

$\frac{2(3x+2)}{(3x+2)(3x-2)} - \frac{2}{3x+1}$

Notice that we have a common factor of (3x + 2) in both the numerator and the denominator of the first fraction. We can cancel these out, but there's one crucial thing to remember: we need to state the restriction that x ≠ -2/3. Why? Because if x were -2/3, the denominator (3x + 2) would be zero, and we can't divide by zero. Always keep an eye out for these restrictions when simplifying algebraic expressions!

Canceling the common factor, we get:

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

Wow, that looks much simpler already, doesn’t it? By canceling out the common factor, we've reduced the complexity of the first fraction. Now, we have two fractions with different denominators: (3x - 2) and (3x + 1). To combine these fractions, we need to find a common denominator. This is like finding a common ground so we can add or subtract them easily. Next up, we'll figure out what that common denominator is and rewrite the fractions accordingly.

Finding a Common Denominator

We've simplified our expression to:

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

Now, we need to find a common denominator so we can subtract these fractions. Remember, to find a common denominator, we look for the least common multiple (LCM) of the denominators. In this case, our denominators are (3x - 2) and (3x + 1). Since these expressions don't share any common factors, their least common multiple is simply their product:

Common Denominator = (3x - 2)(3x + 1)

Now, we need to rewrite each fraction with this common denominator. To do this, we multiply the numerator and denominator of each fraction by the factor that’s missing from its current denominator. For the first fraction, we need to multiply by (3x + 1), and for the second fraction, we need to multiply by (3x - 2). Let's do that:

$\frac{2}{3x-2} \cdot \frac{3x+1}{3x+1} - \frac{2}{3x+1} \cdot \frac{3x-2}{3x-2}$

This gives us:

$\frac{2(3x+1)}{(3x-2)(3x+1)} - \frac{2(3x-2)}{(3x+1)(3x-2)}$

Now both fractions have the same denominator, so we can combine them. We’re getting closer to our final simplified fraction! The next step is to actually combine the numerators and see what we get.

Combining the Fractions

We've reached the point where our expression looks like this:

$\frac{2(3x+1)}{(3x-2)(3x+1)} - \frac{2(3x-2)}{(3x+1)(3x-2)}$

Since the denominators are now the same, we can combine the numerators. This means subtracting the second numerator from the first. Let’s do it carefully, making sure to distribute the negative sign correctly:

$\frac{2(3x+1) - 2(3x-2)}{(3x-2)(3x+1)}$

Now, let's expand the numerators:

$\frac{6x + 2 - (6x - 4)}{(3x-2)(3x+1)}$

Distribute the negative sign in the second term:

$\frac{6x + 2 - 6x + 4}{(3x-2)(3x+1)}$

Now, we can simplify the numerator by combining like terms. Notice that the 6x and -6x terms cancel each other out. This is a good sign, as it means we're making progress towards simplification:

$\frac{2 + 4}{(3x-2)(3x+1)}$

$\frac{6}{(3x-2)(3x+1)}$

We've simplified the numerator to just 6. Now, let's take a look at our fraction. Can we simplify it any further? It looks like the numerator and denominator don't share any common factors, so we might be at the final step!

Final Simplified Form

We've worked our way through the expression and arrived at:

$\frac{6}{(3x-2)(3x+1)}$

Now, the question is: can we simplify this any further? Looking at the numerator and the denominator, we can see that they don't share any common factors. The numerator is simply 6, and the denominator is the product of (3x - 2) and (3x + 1). There's no way to cancel anything out, so this is indeed our simplest form!

Therefore, the simplified form of the expression $\frac{2(3x+2)}{9x^2-4} - \frac{2}{3x+1}$ is:

$\frac{6}{(3x-2)(3x+1)}$

And that’s it, guys! We’ve successfully simplified the expression into a single fraction in its simplest form. Remember, the key steps were factoring, finding a common denominator, combining the fractions, and simplifying the result. Always keep an eye out for common factors and remember to state any restrictions on x. Great job, and I hope you found this helpful!

Conclusion

Simplifying algebraic expressions might seem challenging at first, but with practice, you'll get the hang of it. Remember the key steps we used today: factoring, finding common denominators, combining fractions, and simplifying. These techniques are crucial not just for math problems but also for various applications in science and engineering. By mastering these skills, you're building a strong foundation for more advanced topics in mathematics. So, keep practicing, keep exploring, and don't be afraid to tackle those complex expressions. You've got this!