Simplifying Algebraic Expressions How To Solve 4x - 2(3x - 9)

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Hey guys! Let's dive into simplifying algebraic expressions, a fundamental concept in mathematics. In this article, we'll break down a specific problem and provide a detailed, step-by-step solution. We'll also explore the underlying principles so you can tackle similar problems with confidence. So, grab your thinking caps, and let's get started!

Understanding Algebraic Expressions

Before we jump into the problem, let's quickly recap what algebraic expressions are all about. Algebraic expressions are combinations of variables (like x, y, or z), constants (numbers), and mathematical operations (addition, subtraction, multiplication, division, etc.). Simplifying these expressions involves rearranging and combining terms to make them easier to understand and work with. Think of it like tidying up a messy room – you want to group similar items together and get rid of any unnecessary clutter.

The key to simplifying algebraic expressions lies in following the order of operations (PEMDAS/BODMAS) and applying the distributive property correctly. The order of operations dictates the sequence in which we perform calculations: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). The distributive property allows us to multiply a term by a group of terms inside parentheses. Mastering these two concepts is crucial for simplifying any algebraic expression.

Why is simplifying expressions so important? Well, simplified expressions are much easier to work with in further calculations, such as solving equations or graphing functions. They also provide a clearer picture of the relationship between variables and constants. Imagine trying to build a house from a blueprint filled with complicated symbols and notations – it would be a nightmare! Simplifying expressions is like translating that blueprint into a clear, concise set of instructions that anyone can follow. So, let's get to the problem at hand and see how these concepts work in practice.

Problem: Simplifying the Expression 4x - 2(3x - 9)

Okay, let's get to the heart of the matter! Our problem is to simplify the expression: 4x - 2(3x - 9). This expression looks a bit complex at first glance, but don't worry, we'll break it down step by step. The goal is to find an equivalent expression from the following options:

  • A. 18 - 2x
  • B. -10x - 18
  • C. -2x - 18
  • D. 10x - 18

To solve this, we'll need to apply the distributive property and combine like terms. Remember, the distributive property states that a(b + c) = ab + ac. We'll use this to get rid of the parentheses in our expression. After that, we'll combine like terms, which are terms that have the same variable raised to the same power (e.g., 4x and -6x are like terms, while 4x and 4x² are not). By following these steps carefully, we can transform the original expression into a simpler, equivalent form. Let’s start the simplification process and see which of the options matches our result.

Step-by-Step Solution

Let’s break down the simplification process step-by-step to make it super clear for everyone. Here's how we can simplify the expression 4x - 2(3x - 9):

Step 1: Apply the Distributive Property

The first thing we need to do is get rid of those parentheses. To do this, we'll use the distributive property. Remember, we need to multiply the -2 outside the parentheses by each term inside the parentheses. So, -2 * (3x - 9) becomes (-2 * 3x) + (-2 * -9). Let's do the multiplication:

  • -2 * 3x = -6x
  • -2 * -9 = 18

So, after applying the distributive property, our expression becomes: 4x - 6x + 18. Notice how the subtraction sign in front of the 2 becomes important when distributing. A common mistake is to only multiply -2 by 3x and forget to multiply it by -9. Always double-check your signs to avoid errors!

Step 2: Combine Like Terms

Now that we've eliminated the parentheses, it's time to combine like terms. In our expression 4x - 6x + 18, the like terms are 4x and -6x. Remember, like terms are terms that have the same variable raised to the same power. To combine them, we simply add or subtract their coefficients (the numbers in front of the variables). In this case, we have 4x - 6x. Subtracting the coefficients, we get 4 - 6 = -2. So, 4x - 6x simplifies to -2x. Our expression now looks like this: -2x + 18.

Step 3: Rearrange the Terms (Optional)

While -2x + 18 is a perfectly valid simplified expression, it's often customary to write the constant term first. This is just a matter of preference and doesn't change the value of the expression. To rearrange the terms, we simply swap their positions. So, -2x + 18 becomes 18 - 2x. And there you have it! We've successfully simplified the expression.

By following these steps carefully, we've transformed the original expression 4x - 2(3x - 9) into its simplified form, 18 - 2x. Now, let's compare our result with the given options and see which one matches.

Identifying the Correct Answer

Okay, we've done the hard work of simplifying the expression. Now comes the satisfying part – finding the correct answer! We simplified 4x - 2(3x - 9) to 18 - 2x. Let's look at the options again:

  • A. 18 - 2x
  • B. -10x - 18
  • C. -2x - 18
  • D. 10x - 18

Comparing our simplified expression (18 - 2x) with the options, it's clear that option A, 18 - 2x, is the correct answer. Woohoo! We nailed it! This illustrates the importance of careful and systematic simplification. By following the order of operations and applying the distributive property correctly, we were able to arrive at the right answer. It's easy to make small mistakes along the way, especially with the signs, so always double-check your work. Let's recap the key takeaways from this problem and discuss some common pitfalls to avoid.

Key Takeaways and Common Pitfalls

Alright, let's recap the main points and talk about some common mistakes to watch out for when simplifying algebraic expressions. This will help solidify your understanding and prevent future slip-ups. Remember, practice makes perfect, so the more you work through these types of problems, the more comfortable you'll become.

Key Takeaways:

  • Order of Operations (PEMDAS/BODMAS): Always follow the correct order of operations. Parentheses/Brackets first, then Exponents/Orders, then Multiplication and Division (from left to right), and finally Addition and Subtraction (from left to right).
  • Distributive Property: This is crucial for eliminating parentheses. Remember to multiply the term outside the parentheses by every term inside the parentheses. Pay close attention to the signs!
  • Combining Like Terms: Only combine terms that have the same variable raised to the same power. Add or subtract their coefficients.
  • Double-Check Your Work: It's always a good idea to go back and review your steps, especially the signs, to avoid careless errors.

Common Pitfalls:

  • Forgetting the Distributive Property: This is a big one! Students often forget to multiply the term outside the parentheses by all the terms inside. Make sure you distribute correctly!
  • Sign Errors: Signs can be tricky. Pay close attention to negative signs, especially when distributing. Remember that a negative times a negative is a positive.
  • Combining Unlike Terms: You can only combine like terms. Don't try to add or subtract terms with different variables or exponents.
  • Ignoring Order of Operations: Skipping steps or performing operations in the wrong order can lead to incorrect results. Always follow PEMDAS/BODMAS.

By keeping these key takeaways and common pitfalls in mind, you'll be well-equipped to tackle any algebraic expression simplification problem that comes your way. Now, let's reinforce our understanding with some additional examples.

Additional Examples

To really master simplifying algebraic expressions, it's helpful to work through a few more examples. Let's try a couple of different problems to see how the same principles apply in various situations. Remember, the key is to break down each problem into manageable steps and apply the rules we've discussed.

Example 1: Simplify 3(2x + 5) - 4x

  1. Distribute: 3 * (2x + 5) = 6x + 15. So, the expression becomes 6x + 15 - 4x.
  2. Combine Like Terms: 6x and -4x are like terms. 6x - 4x = 2x. So, the expression simplifies to 2x + 15.

Example 2: Simplify -2(x - 3) + 5(2x + 1)

  1. Distribute: -2 * (x - 3) = -2x + 6 and 5 * (2x + 1) = 10x + 5. So, the expression becomes -2x + 6 + 10x + 5.
  2. Combine Like Terms: -2x and 10x are like terms, and 6 and 5 are like terms. -2x + 10x = 8x and 6 + 5 = 11. So, the expression simplifies to 8x + 11.

Example 3: Simplify 4x - (2x + 7)

  1. Distribute: Think of the minus sign in front of the parentheses as a -1 being multiplied. -1 * (2x + 7) = -2x - 7. So, the expression becomes 4x - 2x - 7.
  2. Combine Like Terms: 4x and -2x are like terms. 4x - 2x = 2x. So, the expression simplifies to 2x - 7.

By working through these examples, you can see how the same principles of distribution and combining like terms apply in different scenarios. The more you practice, the more confident you'll become in your ability to simplify algebraic expressions. Remember to always pay close attention to the signs and follow the order of operations. Now, let's wrap up with a final summary and some encouragement for your continued learning.

Conclusion: Keep Practicing!

Great job, everyone! We've covered a lot in this article about simplifying algebraic expressions. From understanding the basic concepts to working through step-by-step solutions and exploring additional examples, you've gained a solid foundation for tackling these types of problems. Remember, simplifying expressions is a fundamental skill in algebra, and it's essential for solving more complex equations and problems later on.

The key to mastering this skill is consistent practice. Don't be discouraged if you make mistakes – that's a natural part of the learning process. Each mistake is an opportunity to learn and improve. Work through as many examples as you can, and don't hesitate to ask for help if you're stuck. There are plenty of resources available online and in textbooks to support your learning.

So, keep practicing, stay curious, and you'll become a pro at simplifying algebraic expressions in no time! You've got this! And remember, math can be fun when you approach it with a positive attitude and a willingness to learn. Keep exploring, keep challenging yourself, and you'll be amazed at what you can achieve. Until next time, happy simplifying!