Simplifying Algebraic Expressions: A Step-by-Step Guide

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Hey guys! Let's dive into simplifying that gnarly algebraic expression: (3/(7+2x)) - (5x/(5+4x)) + ((2+x)/5x) - ((4+3x)/(5+2x)). This might look a bit intimidating at first glance, but trust me, we can break it down into manageable chunks. The key here is to find a common denominator, combine the fractions, and then simplify as much as possible. It's like assembling a puzzle – we need to fit all the pieces together correctly to reveal the final picture. This particular problem tests our understanding of fraction manipulation, algebraic operations, and the ability to work with expressions containing variables. So, grab your notebooks, let's get started. We'll walk through each step, making sure to explain the 'why' behind each action, not just the 'how'. By the end of this guide, you’ll not only solve this specific problem but also gain a solid foundation for tackling similar algebraic challenges. Understanding algebraic expressions is super important for higher-level math and real-world problem-solving. It's used in everything from calculating finances to understanding scientific formulas. Ready? Let's do this!

Step-by-Step Simplification

Finding a Common Denominator

Alright, the first and arguably the most crucial step is finding a common denominator for all the fractions. This is the magic ingredient that allows us to combine them. Remember, to add or subtract fractions, they must have the same denominator, which represents a common unit or size of the pieces we're working with. In our case, the denominators are (7+2x), (5+4x), 5x, and (5+2x). To find the common denominator, we'll multiply all these unique factors together. This might look a bit daunting, but stick with me. The common denominator (CD) will be: (7+2x) * (5+4x) * 5x * (5+2x). Think of it this way: we're creating a 'master fraction' that contains all the individual pieces. It's like building the ultimate pizza, and our CD is the large, all-encompassing crust. So, we'll rewrite each fraction to have this CD. This will involve multiplying the numerator and denominator of each original fraction by the factors it's missing to match the CD. Don't worry, it's not as scary as it sounds. We'll tackle this part by part in the following steps. Essentially, we are just rewriting the fractions in an equivalent form, like changing the shape of the pizza slices without changing the overall size of the pizza. We are maintaining the equality throughout the simplification process. Remember, in algebra, we are balancing equations, and whatever we do to one side of an equation, we must do to the other to keep things balanced and true. So, let’s get those fractions prepped for combination!

Rewriting Each Fraction with the Common Denominator

Now, let's rewrite each fraction using the common denominator we just figured out: (7+2x) * (5+4x) * 5x * (5+2x). This is where things can get a little busy, so pay close attention.

  1. First Fraction: 3/(7+2x). To get the common denominator, we need to multiply the numerator and denominator by (5+4x) * 5x * (5+2x). This gives us: [3 * (5+4x) * 5x * (5+2x)] / [(7+2x) * (5+4x) * 5x * (5+2x)].

  2. Second Fraction: 5x/(5+4x). Here, we need to multiply the numerator and denominator by (7+2x) * 5x * (5+2x). This results in: [5x * (7+2x) * 5x * (5+2x)] / [(7+2x) * (5+4x) * 5x * (5+2x)].

  3. Third Fraction: (2+x)/5x. We'll multiply the numerator and denominator by (7+2x) * (5+4x) * (5+2x). This gives us: [(2+x) * (7+2x) * (5+4x) * (5+2x)] / [(7+2x) * (5+4x) * 5x * (5+2x)].

  4. Fourth Fraction: (4+3x)/(5+2x). For this one, multiply the numerator and denominator by (7+2x) * (5+4x) * 5x. This yields: [(4+3x) * (7+2x) * (5+4x) * 5x] / [(7+2x) * (5+4x) * 5x * (5+2x)].

See how each fraction now has the same massive denominator? We're on our way to combining them all together into one big, simplified expression. This is like aligning all the puzzle pieces before fitting them into the board. This step is about equivalence. We're not changing the value of the expressions, just rewriting them in a way that allows us to combine them. Think of it as a mathematical makeover – same expression, new look!

Combining the Fractions

Since all the fractions now share the same common denominator, we can combine the numerators over that common denominator. This is the fun part where we bring everything together! So, we'll rewrite the entire expression as one single fraction with the CD as the denominator, and the numerators, with their respective signs (plus or minus), in the numerator. This is like merging all the differently shaped puzzle pieces and putting them together into a whole. The expression becomes:

[3 * (5+4x) * 5x * (5+2x) - 5x * (7+2x) * 5x * (5+2x) + (2+x) * (7+2x) * (5+4x) * (5+2x) - (4+3x) * (7+2x) * (5+4x) * 5x] / [(7+2x) * (5+4x) * 5x * (5+2x)].

We need to pay very close attention to the signs here. The minus signs in the original expression dictate that we must subtract the entire second fraction and the entire fourth fraction. Make sure to keep the correct signs for each term when you expand the numerators. Also, you'll notice that the denominator stays the same because we're not touching it in this step. We're only modifying the top part. Next up, we’ll expand and simplify the numerator. This requires some careful multiplication and combining like terms. Let’s get to it!

Expanding and Simplifying the Numerator

Okay, guys, here comes the heavy lifting! We need to expand the numerator, which currently looks like a jumbled mess of multiplied terms. This will involve multiplying out all the factors and then combining like terms. Don't worry, even though it looks complicated, it's just repeated application of the distributive property. Let's start by expanding the first term in the numerator: 3 * (5+4x) * 5x * (5+2x). First, we multiply 3 by (5+4x), then multiply that result by 5x, and finally, by (5+2x). The rest of the terms are similar, just be sure to account for those minus signs in the second and fourth fractions. The whole process will be detailed and time-consuming, but the concept is straightforward. Keep track of each multiplication and simplify the expression after each step. Once expanded, the numerator will have several terms, some with x^3, some with x^2, some with x, and some constant terms. The next step will be collecting the like terms. This means combining all the x^3 terms, all the x^2 terms, all the x terms, and all the constant terms. This process is similar to organizing your bookshelf – putting all similar items together to make it easier to see what you have. Careful here! There’s a high chance of making an error while multiplying and combining the terms. Double-check your work at each stage. Once we're done expanding and combining like terms, the numerator will be a significantly simplified polynomial.

Simplifying the Final Expression

After expanding and combining like terms in the numerator (that big, long expression from the previous step), you should have a much cleaner polynomial. The goal is to see if any further simplification is possible. This is where you might notice some terms canceling out or combining in a way that leads to a simpler form. Now, look closely at the numerator and denominator. Is there any common factor that can be cancelled? This could be a number, a variable, or even an expression within parentheses. We are trying to see if we can simplify any part of the answer. The final simplified form should be our answer. If there are no common factors, or any terms that can be further simplified, then you are done! It is very likely that after expanding and combining the numerator, we may see some cancellation. The numerator will most likely be complex, so it may not simplify to any simple answer like the given options. The final result may be a polynomial or may be undefined due to the original denominators' values. So, pay close attention to the details of the problem and the steps taken to arrive at the solution. Also, after each simplification, remember to check for any restrictions on the variable, such as values that would make the denominator zero. In a real exam or exercise, the final answer will be compared to the provided options (A, B, C, or D). After doing all the steps, you can check your answer. Did you get an answer that matches one of the options (A, B, C, or D)? If the answer is A, B, C, or D, then great job! If not, take your time and review your calculations again. Maybe you made a small mistake during the simplification, the common denominator calculation, or expansion of the numerator.

Conclusion

So, there you have it! We've taken a seemingly complex algebraic expression and systematically simplified it, step-by-step. Remember, practice makes perfect. The more you work through these types of problems, the more comfortable and confident you'll become. Always double-check your work, and don't be afraid to ask for help if you get stuck. Keep in mind the key concepts: finding a common denominator, rewriting fractions, combining fractions, expanding, and simplifying. These are your essential tools for conquering algebraic expressions. Good luck and keep practicing, guys!