Expand -4(-2/5 + 3x) Using The Distributive Property

Hey guys, let's dive into the awesome world of algebra and tackle a problem that'll make your math brains tingle! We're gonna break down how to use the distributive property to expand an expression. This is a super handy tool in your math arsenal, and once you get the hang of it, you'll be expanding expressions like a pro. Today's challenge is to expand -4ig(-rac{2}{5}+3xig) and figure out which of the given options is the equivalent expression.

Now, what exactly is the distributive property? Think of it like sharing the wealth. When you have a number outside a set of parentheses, like our $-4$ here, it needs to be multiplied by every single term inside those parentheses. It's like that one friend who always brings snacks to the party – they share with everyone! In our case, the $-4$ needs to be distributed to both -rac{2}{5} and $3x$. So, we'll multiply $-4$ by -rac{2}{5}, and then we'll multiply $-4$ by $3x$. This is the core idea behind expanding expressions using the distributive property. It allows us to get rid of the parentheses and work with a simpler, more expanded form of the expression. It's a foundational concept in algebra that pops up everywhere, from solving equations to simplifying complex expressions. Understanding it thoroughly will make tackling more advanced math concepts a breeze. We're going to go step-by-step, so even if this seems a bit tricky at first, stick with me, and we'll get through it together. Remember, practice makes perfect, and by working through this example, you're already on your way to mastering the distributive property.

So, let's get down to business with our expression: -4ig(-rac{2}{5}+3xig). The first step in applying the distributive property is to identify the number outside the parentheses and the terms inside. Here, our number outside is $-4$, and the terms inside are -rac{2}{5} and $3x$. We need to multiply $-4$ by each of these terms.

First, let's multiply $-4$ by the fraction -rac{2}{5}. Remember, multiplying two negative numbers results in a positive number. So, we have:

-4 imes -rac{2}{5}

To multiply a whole number by a fraction, you can think of the whole number as a fraction with a denominator of 1:

-rac{4}{1} imes -rac{2}{5}

Now, multiply the numerators together and the denominators together:

rac{(-4) imes (-2)}{1 imes 5} = rac{8}{5}

Great job! That's the first part of our expanded expression. We've successfully distributed the $-4$ to the first term inside the parentheses, and the result is rac{8}{5}. This is a positive fraction because, as we noted, a negative multiplied by a negative always yields a positive. This sign change is a crucial detail to watch out for when applying the distributive property, especially when negative numbers are involved.

Next, we need to distribute the $-4$ to the second term inside the parentheses, which is $3x$. So, we multiply:

$-4 imes 3x$

When multiplying a number by a term with a variable, you simply multiply the numerical coefficients. The variable remains the same. So:

$-4 imes 3x = -12x$

And there you have it! The second part of our expanded expression is $-12x$. This is a negative term because we multiplied a negative number ($-4$) by a positive number ($3x$). Again, pay close attention to the signs! This step shows how the distributive property works with variables; it allows us to combine the constant factor with the coefficient of the variable term, simplifying the expression further.

Now, we combine the results of both multiplications. We found that -4 imes -rac{2}{5} = rac{8}{5} and $-4 imes 3x = -12x$. So, the fully expanded expression is:

rac{8}{5} - 12x

This is our final answer after applying the distributive property. We've successfully removed the parentheses and have a simplified expression. This process is fundamental in algebra, and it's important to be comfortable with it. Each step, from handling fractions to managing negative signs and variables, builds upon basic arithmetic rules, demonstrating how interconnected mathematical concepts are. Keep practicing these steps, and you'll find that problems like this become second nature.

Now, let's look at the options provided to see which one matches our result:

A. -rac{8}{5}+12 x B. -rac{8}{5}-12 x C. rac{8}{5}+12 x D. rac{8}{5}-12 x

Comparing our expanded expression, rac{8}{5} - 12x, with the options, we can see that Option D is the correct equivalent expression. It perfectly matches our calculated result, including both the numerical term and the variable term with their correct signs. This confirms that our application of the distributive property was accurate. It's always a good idea to double-check your work, especially with signs, as a small error can lead to a completely different answer. We multiplied a negative by a negative to get a positive for the first term, and a negative by a positive to get a negative for the second term, which is exactly what Option D reflects.

To recap, the distributive property is a powerful algebraic tool that lets us simplify expressions by multiplying a factor outside parentheses by each term inside. When we applied it to -4ig(-rac{2}{5}+3xig), we first multiplied $-4$ by -rac{2}{5}, which gave us rac{8}{5} (a positive result because negative times negative is positive). Then, we multiplied $-4$ by $3x$, which resulted in $-12x$ (a negative result because negative times positive is negative). Putting these together, we get rac{8}{5} - 12x. This methodical approach ensures accuracy. Remember, mastering these fundamental algebraic manipulations is key to success in higher-level mathematics. So, keep practicing, keep asking questions, and keep exploring the fascinating world of math. You've got this!

It's also worth noting how crucial attention to detail is in mathematics, especially with signs. A single misplaced negative sign can completely alter the outcome of an expression. In this problem, we encountered two multiplications involving negative numbers. The first, -4 imes -rac{2}{5}, resulted in a positive rac{8}{5}. This is because when two negative numbers are multiplied, the product is always positive. This is a fundamental rule of integer arithmetic that extends to rational numbers and algebraic expressions. If we had mistakenly thought negative times negative is negative, we would have arrived at option A or B, which are incorrect. The second multiplication, $-4 imes 3x$, resulted in $-12x$. Here, a negative number was multiplied by a positive number, and the rule is that the product of a negative and a positive is always negative. If we had messed up this sign, we might have ended up with option C, which has a positive $12x$. Therefore, understanding and correctly applying the rules of signs is paramount when working with the distributive property and any algebraic manipulation. It’s the bedrock upon which complex calculations are built. Practicing these rules consistently will build your confidence and accuracy. Keep your eye on those signs, guys!

Let's think about the structure of the expression and how the distributive property maintains its value. The original expression, -4ig(-rac{2}{5}+3xig), represents a single value. The distributive property essentially rewrites this value in an equivalent form without changing its overall worth. The terms inside the parentheses, -rac{2}{5} and $3x$, represent two parts of a whole. When we multiply the entire sum by $-4$, we're scaling that whole sum by $-4$. By distributing, we're scaling each individual part by $-4$ and then adding those scaled parts together. The distributive property guarantees that these two methods yield the same result. This equivalence is the essence of algebraic manipulation – transforming expressions into different forms while preserving their underlying value. This concept is fundamental to solving equations, where we often manipulate expressions on either side of the equals sign to isolate variables. The distributive property is one of the key tools that allows us to perform these manipulations correctly. It’s a principle that underpins much of what we do in algebra, ensuring that our steps are mathematically sound and that the solutions we find are valid. So, when you see a number outside parentheses, remember it's not just a multiplication; it's an application of a fundamental principle that ensures mathematical equivalence. It’s like having a secret decoder ring for algebraic expressions!

Finally, let's briefly consider why the other options are incorrect. Option A, -rac{8}{5}+12 x, would imply that -4 imes -rac{2}{5} resulted in -rac{8}{5} and $-4 imes 3x$ resulted in $12x$. Both of these are incorrect sign-wise. Option B, -rac{8}{5}-12 x, correctly gets the second term's sign but incorrectly gets the first term's sign. This indicates a misunderstanding of multiplying two negative numbers. Option C, rac{8}{5}+12 x, correctly gets the first term's sign but incorrectly gets the second term's sign. This shows an error in multiplying a negative by a positive. Our derived expression rac{8}{5} - 12x correctly addresses both multiplications, confirming D as the sole correct answer. Understanding why the other options are wrong helps solidify your understanding of the rules being applied. It's like learning to drive – you don't just learn how to steer straight; you also learn what happens if you turn the wheel too sharply or brake too late. By analyzing the incorrect options, we reinforce the correct method. Keep up the great work, everyone!