Solving The Equation 0.25[2.5x + 1.5(x-4)] = -x

Hey guys! Today, we're diving into the exciting world of algebra to solve the equation $0.25[2.5x + 1.5(x-4)] = -x$. This might look a bit intimidating at first, but don't worry, we'll break it down step-by-step so it's super easy to understand. We'll cover each part of the equation, making sure to simplify and clarify everything as we go. Solving equations like this is a fundamental skill in mathematics, and mastering it will definitely help you in future math endeavors. So, grab your pencils and let's get started!

Step-by-Step Solution

1. Distribute Inside the Parentheses

Okay, first things first, let's tackle what's inside the parentheses. We need to distribute the 1.5 across the (x - 4) term. What does that mean? It simply means we multiply 1.5 by both x and -4. This is a crucial step in simplifying the equation and making it more manageable.

So, let's break it down:

• 1. 5 times x equals 1.5x.
• 2. 5 times -4 equals -6.

So, $1.5(x - 4)$ becomes $1.5x - 6$. Now we can rewrite the equation as:

$0. 25[2.5x + 1.5x - 6] = -x$

This might seem like a small change, but it's a significant step towards simplifying the entire equation. Distributing properly is key to solving more complex algebraic problems, so it's a great skill to practice and perfect. Remember, math is like building blocks – each step builds on the previous one, so getting this foundational step right is super important. Keep this in mind as we move forward to the next step. Let's keep going!

2. Combine Like Terms Inside the Brackets

Alright, now that we've handled the parentheses, let's focus on what's inside the brackets. Notice that we have two terms with 'x' in them: 2.5x and 1.5x. These are what we call "like terms," and we can combine them to simplify things further. Combining like terms is like putting all similar things together – in this case, we're adding the two 'x' terms to make our equation cleaner and easier to work with.

So, let's do it:

• Add 2.5x and 1.5x. What do we get? 4x!

Now our equation looks like this:

$0. 25[4x - 6] = -x$

See how much simpler that looks already? By combining like terms, we've reduced the clutter and made the equation more straightforward. This step is all about organization and making sure we're working with the most simplified form of the expression. Remember, simplification is your friend in math! It helps prevent errors and makes the problem much easier to solve. Next up, we'll tackle that 0.25 outside the brackets. Keep going, you're doing great!

3. Distribute the 0.25

Okay, next up we need to get rid of those brackets. To do this, we'll distribute the 0.25 across the terms inside the brackets – that's the 4x and the -6. Distributing is just a fancy way of saying we're going to multiply 0.25 by each of these terms. This is a crucial step in unwrapping the equation and isolating our 'x' terms. Let's break it down piece by piece to make sure we get it spot on.

• First, we multiply 0.25 by 4x. What's 0.25 times 4? It's 1! So, 0.25 times 4x is simply 1x, which we usually just write as x.
• Next, we multiply 0.25 by -6. What's 0.25 times -6? It's -1.5.

Now, let's put it all together. Our equation now looks like this:

$x - 1.5 = -x$

Doesn't that look so much simpler? Distributing the 0.25 has really cleaned things up. Remember, each step we take is about making the equation more manageable and getting us closer to our final answer. Now that we've distributed, the next step is to gather all our 'x' terms on one side of the equation. Let's keep the momentum going!

4. Move the -x Term to the Left Side

Now, let's get all the 'x' terms together on one side of the equation. Currently, we have an 'x' on the left side and a '-x' on the right side. To get them together, we need to move that '-x' from the right to the left. How do we do that? Simple! We perform the opposite operation. Since it's a negative 'x' on the right, we'll add 'x' to both sides of the equation. This keeps the equation balanced, which is super important in algebra.

So, let’s add 'x' to both sides:

$x - 1.5 + x = -x + x$

On the left side, we have 'x' plus 'x', which gives us 2x. On the right side, '-x' plus 'x' cancels out, leaving us with zero. Our equation now looks like this:

$2x - 1.5 = 0$

We're getting closer to solving for 'x'! By moving the '-x' term, we've simplified the equation even further. Remember, our goal is to isolate 'x' on one side, so every step we take should be in that direction. Now that we have all our 'x' terms on the left, it's time to get rid of that -1.5. Let's see how to do that in the next step. Keep it up, you're doing awesome!

5. Add 1.5 to Both Sides

Alright, let's keep isolating 'x'. We've got $2x - 1.5 = 0$, and we need to get that '-1.5' out of the way. Just like before, we'll do the opposite operation to move it. Since we're subtracting 1.5 on the left side, we'll add 1.5 to both sides of the equation. This keeps everything balanced and helps us move closer to solving for 'x'.

So, let's add 1.5 to both sides:

$2x - 1.5 + 1.5 = 0 + 1.5$

On the left side, -1.5 plus 1.5 cancels out, leaving us with just 2x. On the right side, 0 plus 1.5 is simply 1.5. Our equation now looks like this:

$2x = 1.5$

See how clean and simple it's becoming? We're almost there! By adding 1.5 to both sides, we've successfully isolated the term with 'x' on one side of the equation. Now, there's just one more step to take to get 'x' all by itself. We need to get rid of that 2 that's multiplying 'x'. Let's tackle that in the next step. You're doing such a fantastic job – let's finish strong!

6. Divide Both Sides by 2

Okay, we're in the home stretch now! We have the equation $2x = 1.5$, and our final step is to get 'x' completely alone on one side. Right now, 'x' is being multiplied by 2. So, to undo that multiplication, we need to do the opposite operation, which is division. We'll divide both sides of the equation by 2. This will keep the equation balanced and finally give us the value of 'x'.

Let's divide both sides by 2:

rac{2x}{2} = rac{1.5}{2}

On the left side, 2x divided by 2 is simply 'x'. On the right side, 1.5 divided by 2 is 0.75. So, our equation simplifies to:

$x = 0.75$

And there we have it! We've solved for 'x'.

Final Answer

So, the solution to the equation $0.25[2.5x + 1.5(x-4)] = -x$ is:

$x = 0.75$

Conclusion

Awesome job, guys! We successfully solved the equation $0.25[2.5x + 1.5(x-4)] = -x$ by breaking it down into manageable steps. We started by distributing, then combined like terms, moved terms around to isolate 'x', and finally solved for 'x'. Remember, the key to tackling these types of problems is to take it one step at a time and stay organized. Each step builds on the previous one, and with a bit of practice, you'll become a pro at solving algebraic equations. Keep up the great work, and remember to apply these skills to other math problems you encounter. You've got this!