Solve -0.4(x+3)=4: Easy Steps
Hey guys, let's dive into a super common math problem that pops up all the time: solving linear equations. Today, we're tackling this specific one: $-0.4(x+3)=4$. Now, I know sometimes equations can look a bit intimidating with those decimals and parentheses, but trust me, once you break it down, it's totally manageable. We're going to walk through this step-by-step, making sure we understand why we're doing each part. This isn't just about getting the answer; it's about building that confidence so you can handle any similar problem that comes your way. We'll be looking at the options provided (A. -13, B. -0.6, C. 1.4, D. 7) to see which one correctly balances our equation. So, grab your favorite beverage, get comfy, and let's unravel this mathematical mystery together!
Understanding the Goal: Isolating 'x'
The main mission when we're asked to find the solution to the equation $-0.4(x+3)=4$ is to figure out what number x needs to be for this statement to be true. In the world of algebra, this means we want to get x all by itself on one side of the equals sign. Think of it like a balancing act. Whatever we do to one side of the equation, we must do to the other side to keep things balanced. Our ultimate goal is to have a result that looks something like x = [some number]. The key is to undo the operations that are currently being applied to x. Right now, x is inside parentheses, being added to 3, and the whole thing is being multiplied by -0.4. Our strategy will be to reverse these operations in the opposite order of how they were applied (think reverse PEMDAS or GEMS β Grouping Symbols, Exponents, Multiplication/Division, Addition/Subtraction).
Step 1: Dealing with the Multiplication
So, we have $-0.4(x+3)=4$. The first thing we need to tackle is that -0.4 that's multiplying the entire expression in the parentheses. To undo multiplication, we use its inverse operation: division. So, we're going to divide both sides of the equation by -0.4. This is a crucial step because it helps us get rid of that coefficient outside the parentheses, bringing us closer to isolating x. Remember, whatever we do to one side, we have to do to the other.
On the left side, when we divide $-0.4(x+3)$ by -0.4, the -0.4 on the top and bottom cancel each other out, leaving us with just $(x+3)$. Now, let's look at the right side. We need to calculate $4 \div -0.4$. Dividing by a decimal can sometimes trip people up, but think of it this way: how many times does 0.4 go into 4? It's the same as asking how many times does 4/10 go into 40/10. Or, more simply, you can think of it as $4 \div (4/10) = 4 imes (10/4) = 10$. Since we are dividing a positive number (4) by a negative number (-0.4), our result will be negative. So, $4 \div -0.4 = -10$.
After this step, our equation now looks much simpler: $(x+3) = -10$. See? We've already made significant progress! The parentheses are gone, and we're just left with a simple addition problem to solve. This is why carefully applying inverse operations is so powerful in algebra. It systematically simplifies the equation until the variable is laid bare.
Step 2: Undoing the Addition
Alright, guys, we're at $(x+3) = -10$. Our goal is still to get x by itself. Right now, x has 3 being added to it. The inverse operation of addition is subtraction. So, to isolate x, we need to subtract 3 from both sides of the equation. This will cancel out the '+3' on the left side.
On the left side, when we subtract 3 from $(x+3)$, we get $x + 3 - 3$, which simplifies to just x. Now, we move to the right side. We have $-10$ and we need to subtract 3 from it. Subtracting a positive number is like moving further down the number line into the negatives. So, $-10 - 3$ equals $-13$.
And there we have it! Our equation is now completely solved, and we've found that $x = -13$. This is our potential solution. It's the number that, when plugged back into the original equation, should make the statement true. Always remember to check your work, especially when decimals or negative numbers are involved, to ensure accuracy. This process of undoing operations is the fundamental way we solve algebraic equations, and it works for pretty much all of them!
Step 3: Checking Our Solution
Now, the most important part, especially in math, is to verify your answer. We found that $x = -13$ is the solution to $-0.4(x+3)=4$. Let's plug -13 back into the original equation and see if it holds true. This step is non-negotiable if you want to be absolutely sure you haven't made any small errors along the way. It's like proofreading your work!
Original equation: $-0.4(x+3)=4$ Substitute $x = -13$:
First, let's solve what's inside the parentheses: $-13 + 3$. When you add a positive number to a negative number, you find the difference between their absolute values and take the sign of the number with the larger absolute value. The difference between 13 and 3 is 10, and since -13 has the larger absolute value, the result is negative. So, $-13 + 3 = -10$.
Now our equation looks like this: $-0.4(-10)=4$.
Next, we perform the multiplication: $-0.4 imes -10$. Remember, a negative number multiplied by a negative number always results in a positive number. So, $0.4 imes 10 = 4$.
Therefore, we get $4 = 4$.
Boom! The left side perfectly equals the right side. This confirms that our solution, $x = -13$, is indeed correct. It's always so satisfying when the numbers match up like this, right? This validation process is key to building confidence in your algebraic skills. It shows you that you can trust your methods and your calculations.
Analyzing the Options
We were given four options to choose from:
A. -13 B. -0.6 C. 1.4 D. 7
Our calculations led us directly to $x = -13$. When we plugged this value back into the original equation, it satisfied the equality, proving it to be the correct solution. Therefore, option A is the correct answer. Let's briefly consider why the other options wouldn't work. If we tried, say, option B, $x = -0.6$, we'd get $-0.4(-0.6 + 3) = -0.4(2.4) = -0.96$, which is definitely not 4. Similarly, for option C, $x = 1.4$, we'd have $-0.4(1.4 + 3) = -0.4(4.4) = -1.76$. And for option D, $x = 7$, we'd get $-0.4(7 + 3) = -0.4(10) = -4$. None of these match the right side of our original equation, which is 4. This highlights the importance of meticulous calculation and verification. Sometimes, small mistakes in arithmetic can lead you to the wrong choice, even if your overall strategy is sound. This is why double-checking, especially the arithmetic parts, is super important. Remember, guys, practice makes perfect, and understanding the why behind each step is what truly solidifies your math skills. Keep practicing, and you'll be solving equations like a pro in no time!
Conclusion: Master Linear Equations!
So there you have it, team! We've successfully navigated the process of solving the linear equation $-0.4(x+3)=4$. We started by understanding our goal: to isolate the variable x. We then systematically applied the inverse operations β division to remove the multiplier and subtraction to remove the added constant β to both sides of the equation. Crucially, we finished by plugging our potential solution, $x = -13$, back into the original equation to verify that it indeed made the statement true. This verification step is vital for ensuring accuracy and building confidence.
We saw that our derived solution, $x = -13$, perfectly matched option A, and we briefly explored why the other options were incorrect. This reinforces the idea that in mathematics, there's often a single correct path and answer, achieved through logical steps and precise calculations. Mastering these linear equation solutions isn't just about passing a test; it's about developing critical thinking and problem-solving skills that are applicable in countless areas of life. Keep practicing these types of problems, and don't be afraid to tackle more complex ones. The more you practice, the more intuitive these steps will become, and you'll find yourself solving equations with ease. Remember, every equation you solve is a step towards becoming a math whiz! Keep up the great work, and happy solving!