Solve $0.25[2.5 X+1.5(x-4)]=-x$

Solve $0.25[2.5 x+1.5(x-4)]=-x$

Hey guys, welcome back to Plastik Magazine! Today, we're diving deep into the world of algebra to tackle a rather juicy equation: $0.25[2.5 x+1.5(x-4)]=-x$. Don't let the decimals and brackets scare you off; we're going to break this down step-by-step, making it as clear as a freshly cleaned beaker. This kind of problem is super common in algebra, whether you're in high school or just brushing up on your math skills. Understanding how to simplify and solve these equations is a fundamental building block for tackling more complex mathematical challenges down the line. So, grab your calculators (or just your brainpower!) and let's get this solved. We'll go through each manipulation, explaining why we're doing it, so you're not just following along but truly understanding the process. Our main goal here is to isolate 'x' and find its value. Ready? Let's get to it!

Step 1: Distribute within the Parentheses

The first thing we need to do is simplify the expression inside the brackets. Remember the order of operations (PEMDAS/BODMAS)? Parentheses come first! Inside our main brackets, we have $2.5x + 1.5(x-4)$. The $1.5$ is multiplying the entire $(x-4)$ term. So, we need to distribute the $1.5$ to both the $x$ and the $-4$. This means we multiply $1.5$ by $x$ and $1.5$ by $-4$.

• $1.5 * x = 1.5x$
• $1.5 * -4 = -6$

Now, substitute these back into the expression inside the brackets: $2.5x + 1.5x - 6$.

See? We've already made the equation look a little cleaner. This step is all about unravelling the nested parts of the equation. It’s like peeling an onion – you just keep removing the layers until you get to the core. The distributive property is a key concept here, allowing us to expand expressions and combine like terms more easily. It's one of those algebraic tools you'll use constantly, so getting comfortable with it is a major win.

Step 2: Combine Like Terms Inside the Brackets

Now that we've distributed, we can combine the 'x' terms inside the brackets. We have $2.5x$ and $1.5x$. When you add these together, you get:

• $2.5x + 1.5x = 4x$

So, the expression inside the brackets simplifies to $4x - 6$. Our equation now looks like this: $0.25[4x - 6] = -x$.

Combining like terms is another cornerstone of simplifying algebraic expressions. It makes the expression more concise and easier to work with. Think of it as tidying up your workspace – the less clutter, the more efficient you can be. We're systematically reducing the complexity of the equation, getting closer and closer to finding that elusive value of 'x'. This methodical approach ensures we don't miss any steps and maintain accuracy. It’s really satisfying to see the equation shrink down, right?

Step 3: Distribute the 0.25

Next, we need to deal with the $0.25$ that's multiplying the entire bracketed expression $[4x - 6]$. We apply the distributive property again. Multiply $0.25$ by $4x$ and $0.25$ by $-6$.

• $0.25 * 4x = 1x$ (or just $x$)
• $0.25 * -6 = -1.5$

So, the left side of the equation becomes $x - 1.5$. Our equation is now significantly simpler: $x - 1.5 = -x$.

This step is crucial because it removes the brackets entirely, making the equation a linear one with 'x' terms on both sides. The ability to distribute a factor across a binomial (an expression with two terms) is a fundamental skill. It's all about understanding how multiplication affects sums and differences. This reduction in complexity is what algebraic problem-solving is all about – transforming a seemingly complicated expression into a manageable form. We're well on our way to isolating 'x' now!

Step 4: Isolate the 'x' Terms

Our current equation is $x - 1.5 = -x$. To solve for 'x', we need to get all the 'x' terms on one side of the equation and all the constant terms on the other. Let's move the $-x$ from the right side to the left side. We do this by adding $x$ to both sides of the equation to maintain balance.

• $(x - 1.5) + x = -x + x$
• $2x - 1.5 = 0$

Now, we want to move the constant term, $-1.5$, to the right side. We do this by adding $1.5$ to both sides of the equation.

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

This is where the magic starts to happen! By strategically adding or subtracting terms from both sides, we're systematically isolating the variable. It’s like playing a game of algebraic chess, where each move is designed to bring us closer to checkmating the variable 'x'. Balancing the equation is key – whatever you do to one side, you must do to the other. This keeps the equality true at every step. We've successfully gathered all our 'x's on one side and the numbers on the other.

Step 5: Solve for 'x'

We're at the final hurdle! Our equation is now $2x = 1.5$. To find the value of a single 'x', we need to divide both sides of the equation by the coefficient of 'x', which is 2.

• $2x / 2 = 1.5 / 2$
• $x = 0.75$

And there you have it! The solution to the equation $0.25[2.5 x+1.5(x-4)]=-x$ is $x = 0.75$.

This last step is about undoing the multiplication. Since 'x' is being multiplied by 2, we perform the inverse operation, which is division. This gives us the value of 'x' itself. It's a beautiful, clean solution derived from a slightly more complex starting point. Bravo!

Verification (Optional but Recommended!)

To be absolutely sure, especially in tests or important assignments, it's always a good idea to verify your answer. Plug $x = 0.75$ back into the original equation and see if both sides are equal.

• Left Side: $0.25[2.5(0.75) + 1.5(0.75 - 4)]$

  • $0.25[1.875 + 1.5(-3.25)]$
  • $0.25[1.875 - 4.875]$
  • $0.25[-3]$
  • $-0.75$

• Right Side: $-x$

  • $-(0.75)$
  • $-0.75$

Since the left side ($-0.75$) equals the right side ($-0.75$), our solution is correct! High five!

Conclusion

So there you have it, mathletes! We've successfully navigated the equation $0.25[2.5 x+1.5(x-4)]=-x$ from its initial form to its elegant solution of $x=0.75$. We used fundamental algebraic properties like the distributive property and the concept of combining like terms, all while maintaining the balance of the equation. Remember, guys, the key to mastering algebra isn't just memorizing steps; it's understanding the why behind each step. Practice these types of problems, and soon you'll be solving equations like a pro. Keep those math brains buzzing, and we'll see you in the next article for more problem-solving fun here at Plastik Magazine!