Hey guys! Ever stumbled upon an algebraic expression that looks like a tangled mess? Don't worry, we've all been there. Today, we're going to untangle one such expression: 0.6(3-2x) - 0.4(1.5x+2). Think of it as a puzzle – a mathematical puzzle! And like any good puzzle, there's a systematic way to solve it. We'll break it down step-by-step, so by the end of this guide, you'll be a pro at simplifying expressions like this one.
Understanding the Basics
Before diving into the nitty-gritty, let's quickly recap some fundamental concepts. When we talk about simplifying an expression, we mean rewriting it in a cleaner, more manageable form. This usually involves two main operations: the distributive property and combining like terms. The distributive property is our key to unlocking the parentheses, while combining like terms helps us tidy up the remaining terms. Let's take a closer look at each of these concepts.
The Distributive Property: Our Key to Unlocking Parentheses
The distributive property is a fundamental concept in algebra that allows us to multiply a single term by multiple terms inside parentheses. It states that for any numbers a, b, and c: a(b + c) = ab + ac. In simpler terms, you multiply the term outside the parentheses by each term inside the parentheses. Think of it like sharing: you're distributing the 'a' to both 'b' and 'c'. This property is crucial for expanding expressions and getting rid of those pesky parentheses that often stand in our way. When we apply the distributive property correctly, we transform a more complex expression into a form that's easier to work with. Imagine the parentheses as a gate, and the distributive property is the key to unlock it and access the terms inside.
For example, consider the expression 2(x + 3). Using the distributive property, we multiply 2 by both x and 3, which gives us 2 * x + 2 * 3, simplifying to 2x + 6. This process eliminates the parentheses and makes the expression simpler. In our main problem, 0. 6(3 - 2x) - 0.4(1.5x + 2), we'll apply this property twice, once for each set of parentheses. Mastering the distributive property is like learning a secret handshake in the world of algebra – it opens doors and makes everything flow smoother!
Combining Like Terms: Tidying Up the Expression
Once we've used the distributive property, we'll often find ourselves with a collection of terms. This is where combining like terms comes in handy. Like terms are terms that have the same variable raised to the same power. For example, 3x and -5x are like terms because they both have the variable 'x' raised to the power of 1. Similarly, constants like 7 and -2 are like terms because they don't have any variables. However, 2x and 2x² are not like terms because they have the same variable ('x') but raised to different powers (1 and 2, respectively).
Combining like terms is like sorting your socks – you group together the ones that are similar. To combine like terms, you simply add or subtract their coefficients (the numbers in front of the variables). For instance, 3x - 5x simplifies to -2x because 3 - 5 = -2. Similarly, 7 - 2 simplifies to 5. By combining like terms, we condense the expression into its simplest form, making it easier to understand and work with. Think of it as decluttering your math problem – you're getting rid of the extra baggage and focusing on the essential components. In our expression, we'll identify and combine the 'x' terms and the constant terms to streamline the expression and arrive at our final answer.
Step-by-Step Solution
Now that we've got a good grasp of the basics, let's tackle our expression 0.6(3-2x) - 0.4(1.5x+2) step-by-step. We'll break it down into manageable chunks, making sure we understand each step before moving on.
Step 1: Apply the Distributive Property
Our first mission is to eliminate the parentheses. To do this, we'll use the distributive property on both parts of the expression. First, let's distribute the 0.6 across the (3 - 2x) term. This means multiplying 0.6 by both 3 and -2x:
- 0.6 * 3 = 1.8
- 0.6 * (-2x) = -1.2x
So, the first part of the expression becomes 1.8 - 1.2x. Now, let's move on to the second part and distribute the -0.4 across the (1.5x + 2) term. Remember to pay close attention to the signs!
- -0.4 * 1.5x = -0.6x
- -0.4 * 2 = -0.8
The second part of the expression becomes -0.6x - 0.8. Now, let's put it all together. Our expression, after applying the distributive property, looks like this: 1.8 - 1.2x - 0.6x - 0.8. We've successfully unlocked the parentheses and are ready for the next step!
Step 2: Identify Like Terms
With the parentheses gone, our expression is now a string of terms. Our next task is to identify the like terms. Remember, like terms are those that have the same variable raised to the same power. Looking at our expression, 1. 8 - 1.2x - 0.6x - 0.8, we can see two types of terms: terms with 'x' and constant terms (numbers without variables). The 'x' terms are -1.2x and -0.6x. The constant terms are 1.8 and -0.8. It's like sorting laundry – we're grouping together the similar items. Identifying like terms is a crucial step because it sets us up for the final simplification, where we'll combine these terms to make our expression as concise as possible.
Step 3: Combine Like Terms
Now comes the satisfying part – combining the like terms! We'll start by combining the 'x' terms: -1.2x and -0.6x. To do this, we simply add their coefficients: -1.2 + (-0.6) = -1.8. So, -1.2x - 0.6x simplifies to -1.8x. Next, we'll combine the constant terms: 1.8 and -0.8. Adding these together, we get 1.8 + (-0.8) = 1.0, which is simply 1. Now, we put the simplified terms together. Our expression, 1. 8 - 1.2x - 0.6x - 0.8, simplifies to -1.8x + 1. And there you have it! We've successfully combined the like terms and arrived at the simplified form of our expression.
Step 4: Write the Final Simplified Expression
We've done the hard work, and now it's time to present our final answer in a clear and concise manner. After applying the distributive property and combining like terms, we've simplified the expression 0.6(3-2x) - 0.4(1.5x+2) to -1.8x + 1. This is the simplified form of the original expression, and it's much easier to understand and work with. Sometimes, you might see the answer written as 1 - 1.8x, which is mathematically equivalent. The order of terms doesn't change the value of the expression, as long as the signs are correct. Whether you write -1.8x + 1 or 1 - 1.8x, you've successfully simplified the expression! Congratulations, you've cracked the code!
Common Mistakes to Avoid
Simplifying expressions can be tricky, and there are a few common pitfalls that students often encounter. Being aware of these mistakes can help you avoid them and ensure you get the correct answer every time.
Forgetting to Distribute to All Terms
One of the most common mistakes is forgetting to distribute the term outside the parentheses to all the terms inside. Remember, the distributive property means you multiply the term outside by each term inside the parentheses. For example, in 2(x + 3), you need to multiply 2 by both x and 3. Forgetting to multiply by one of the terms will lead to an incorrect answer. Always double-check that you've distributed correctly to every term within the parentheses. It's like making sure everyone gets a slice of pizza – you wouldn't want to leave anyone out!
Incorrectly Combining Unlike Terms
Another frequent error is trying to combine terms that are not alike. Remember, like terms have the same variable raised to the same power. You can combine 3x and -5x because they both have 'x' to the power of 1, but you cannot combine 2x and 2x² because the powers of 'x' are different. It's like trying to mix apples and oranges – they're both fruits, but they can't be combined in the same way. Only combine terms that are truly alike, and your expressions will stay accurate.
Sign Errors
Sign errors are sneaky little devils that can easily trip you up. When distributing a negative number, be extra careful to apply the negative sign correctly. For instance, in -2(x - 3), you need to distribute -2 to both x and -3, resulting in -2x + 6. A common mistake is to forget that multiplying two negatives results in a positive. Always double-check your signs, especially when dealing with negative numbers. It's like watching your step on a slippery slope – a little extra caution can prevent a fall!
Order of Operations
Sometimes, the order of operations (PEMDAS/BODMAS) can be a source of confusion. Remember to perform multiplication and division before addition and subtraction. If you have an expression like 2 + 3 * x, you need to multiply 3 by x first, and then add 2. Ignoring the order of operations can lead to a completely wrong answer. Keep PEMDAS/BODMAS in mind as your guiding principle, and you'll navigate complex expressions with confidence.
Practice Makes Perfect
The best way to master simplifying expressions is through practice, practice, practice! The more you work through different examples, the more comfortable you'll become with the process. Start with simpler expressions and gradually move on to more complex ones. Try working through similar problems to the one we solved today. You can also find plenty of practice problems online or in textbooks. Each problem you solve is like a step forward on your path to algebraic mastery. So, roll up your sleeves, grab a pencil, and start practicing. You'll be simplifying expressions like a pro in no time!
Conclusion
Simplifying expressions might seem daunting at first, but with a clear understanding of the basics and a step-by-step approach, it becomes much more manageable. We've walked through the process of simplifying 0.6(3-2x) - 0.4(1.5x+2), highlighting the importance of the distributive property, combining like terms, and avoiding common mistakes. Remember, guys, math is like any skill – it gets easier with practice. So, keep practicing, stay curious, and you'll conquer those algebraic expressions in no time! Now, go forth and simplify!