Solving Inequalities: Your First Step

Hey guys! Ever stare at an inequality and just freeze, wondering where to even begin? We've all been there. Today, we're diving deep into the nitty-gritty of solving inequalities, specifically tackling that crucial first step. Our focus inequality is $-4(3-5 x) oun{ extgreater or equal to } -6 x+9$. Getting this initial move right is super important because it sets the stage for the rest of the problem. Mess it up, and your whole solution could go sideways faster than you can say "math whiz." So, let's break down why one particular first step is the correct one and how it leads us down the path to the right answer. We'll be looking at distributing that $-4$ across the terms inside the parentheses, $(3-5x)$. This is a fundamental algebraic operation, and in the context of inequalities, it's just as vital as in equations. Remember, when you multiply or divide an inequality by a negative number, you have to flip the inequality sign. That's a classic trap, and we'll be super careful to avoid it. But for this very first step, we're just focusing on the distribution. We need to multiply $-4$ by $3$ and then multiply $-4$ by $-5x$. The result of $-4$ times $3$ is $-12$. Now, for the second part, $-4$ times $-5x$ gives us a positive $20x$. So, the left side of our inequality, $-4(3-5x)$, becomes $-12+20x$. The right side, $-6x+9$, remains unchanged for this initial step. Therefore, the inequality transforms from $-4(3-5 x) oun{ extgreater or equal to } -6 x+9$ to $-12+20 x oun{ extgreater or equal to } -6 x+9$. This means option D is our correct first step. It's all about carefully applying the distributive property and keeping everything else the same until we're ready for the next stage of solving. We're not flipping any signs yet because we haven't performed any operation that requires it. This careful, step-by-step approach is the hallmark of successful problem-solving, especially in math. Stick with us, and we'll walk through the rest of the solution, making sure you feel confident and ready to tackle any inequality that comes your way. Remember, mastering these foundational steps is key to unlocking more complex algebraic concepts. Let's keep the momentum going and nail this inequality problem together! It’s all about building that solid understanding, one step at a time, so you guys can conquer any math challenge.

The Anatomy of an Inequality

Before we dive headfirst into solving, let's take a moment to appreciate what we're dealing with: an inequality. Unlike equations that state two things are equal, inequalities tell us that two things are not necessarily equal, but rather related by a comparison. This comparison can be greater than ($>$), less than ($<$), greater than or equal to ($oun{ extgreater or equal to }$), or less than or equal to ($oun{ extless or equal to }$). In our specific problem, $-4(3-5 x) oun{ extgreater or equal to } -6 x+9$, we're working with the "greater than or equal to" sign. This little symbol is crucial; it dictates the range of possible solutions, which will be a set of numbers, not just a single value (like in many equations). Think of it like this: an equation is like saying "This door is locked." An inequality is more like saying, "This door is either locked, or it's unlocked, but you can't go through it right now." It opens up a world of possibilities! Our main goal when solving inequalities is to isolate the variable, just like in equations. We want to get 'x' all by itself on one side of the inequality sign. However, the rules for manipulating inequalities have a special twist. While most operations (like adding or subtracting the same value from both sides, or multiplying/dividing by a positive number) are the same as with equations, there's one major exception. If you multiply or divide both sides of an inequality by a negative number, you must reverse the direction of the inequality sign. So, a $>$ becomes a $<$, and a $oun{ extgreater or equal to }$ becomes a $oun{ extless or equal to }$. This rule is absolutely non-negotiable and is a common stumbling block for many students. It's like a secret handshake for inequality algebra! That's why the very first step in our problem, $-4(3-5 x) oun{ extgreater or equal to } -6 x+9$, is so important. We need to simplify the expression on the left side without violating any of these rules. The expression $-4(3-5x)$ involves multiplication, specifically the distribution of $-4$ to both $3$ and $-5x$. This operation itself doesn't involve multiplying or dividing the entire inequality by a negative number; it's simply simplifying one side. Therefore, we perform the distribution: $-4 imes 3 = -12$ and $-4 imes (-5x) = +20x$. So, the left side becomes $-12 + 20x$. The inequality now reads $-12 + 20x oun{ extgreater or equal to } -6x + 9$. This is why option D, $-12+20 x oun{ extgreater or equal to } -6 x+9$, correctly represents the first step. It's a careful application of the distributive property without any premature sign flipping. Understanding these fundamental building blocks makes tackling more complex problems much more manageable. It's all about building that strong foundation, guys!

Demystifying the First Step: Distribution Done Right

Alright folks, let's get back to our star inequality: $-4(3-5 x) oun{ extgreater or equal to } -6 x+9$. The absolute first thing we need to do is simplify the left-hand side. Why? Because it's messy! We've got parentheses and multiplication staring us down. The standard algebraic procedure here is to apply the distributive property. This means we take the number outside the parentheses, which is $-4$ in this case, and multiply it by each term inside the parentheses. So, we've got two multiplication operations to perform: $-4$ multiplied by $3$, and $-4$ multiplied by $-5x$. This is not the step where we start worrying about flipping inequality signs. We are simply simplifying one side of the existing inequality. The inequality sign itself, $oun{ extgreater or equal to }$, remains unchanged for now. Let's do the math, step-by-step: First term: $-4 imes 3$. When you multiply a negative number by a positive number, the result is always negative. So, $-4 imes 3 = -12$. Second term: $-4 imes (-5x)$. Here, we're multiplying two negative numbers. Remember, a negative times a negative equals a positive! So, $-4 imes -5x = +20x$. Putting these two results together, the left side of our inequality, $-4(3-5x)$, is correctly rewritten as $-12 + 20x$. The right side of the inequality, $-6x+9$, is untouched in this initial step. So, the inequality now reads: $-12 + 20x oun{ extgreater or equal to } -6x + 9$. Now, let's look at our options. We're searching for the representation that shows exactly this transformation. Option A gives $-12-20 x oun{ extless or equal to } -6 x+9$. That's wrong because the $-20x$ should be $+20x$, and the sign is flipped incorrectly. Option B gives $-12-20 x oun{ extgreater or equal to } -6 x+9$. Still wrong because of the $-20x$. Option C gives $-12+20 x oun{ extless or equal to } -6 x+9$. This has the correct distribution ($-12+20x$), but it incorrectly flips the inequality sign from $oun{ extgreater or equal to }$ to $oun{ extless or equal to }$. Option D gives $-12+20 x oun{ extgreater or equal to } -6 x+9$. Bingo! This matches our calculated first step perfectly. It correctly applies the distributive property, resulting in $-12+20x$ on the left, while keeping the inequality sign $oun{ extgreater or equal to }$ and the right side $-6x+9$ as they were. This meticulous attention to detail in the first step prevents cascading errors. It’s like building a house – you need a solid foundation before you start putting up walls. Mastering this initial distribution is fundamental to building confidence and accuracy in solving all sorts of algebraic problems, not just inequalities. Keep practicing this, and you’ll be a pro in no time, guys!

Why Other Options Don't Make the Cut

Let's be real, guys. When you're faced with a math problem, especially one involving inequalities, it's easy to get tripped up. The key is to understand the rules and apply them methodically. We've already established that the correct first step for $-4(3-5 x) oun{ extgreater or equal to } -6 x+9$ is to distribute the $-4$ across the terms $(3-5x)$. This yields $-12 + 20x$ on the left side, keeping the inequality sign $oun{ extgreater or equal to }$ and the right side $-6x+9$ intact. So, the correct transformation is $-12+20 x oun{ extgreater or equal to } -6 x+9$. Now, let's dissect why the other options are incorrect. This will help reinforce why our chosen first step is the right one.

• Option A: $-12-20 x oun{ extless or equal to } -6 x+9$ This option gets two things wrong. Firstly, the distribution of $-4$ by $-5x$ should result in a positive $20x$, not a negative $-20x$. So, $-12-20x$ is an incorrect simplification of $-4(3-5x)$. Secondly, it flips the inequality sign from $oun{ extgreater or equal to }$ to $oun{ extless or equal to }$. There has been no operation performed yet that would necessitate flipping the sign. This option shows a misunderstanding of both the distributive property and the rules for manipulating inequalities. It's a double whammy of errors, unfortunately.

• Option B: $-12-20 x oun{ extgreater or equal to } -6 x+9$ Option B corrects the inequality sign, keeping it as $oun{ extgreater or equal to }$, which is good for a first step. However, it still suffers from the same error as Option A regarding the distribution: it incorrectly shows $-20x$ instead of $+20x$. The multiplication of $-4$ by $-5x$ must yield a positive result. Therefore, this option also fails to correctly represent the initial distribution.

• Option C: $-12+20 x oun{ extless or equal to } -6 x+9$ Option C gets the distribution right! It correctly shows $-12+20x$ on the left side, which is exactly what we expect from distributing $-4$ to $(3-5x)$. However, it incorrectly flips the inequality sign. The original inequality is $oun{ extgreater or equal to }$, and for this initial distribution step, there's no reason to change it. Flipping the sign only occurs when you multiply or divide the entire inequality by a negative number. Since we've only simplified one side, the sign should remain the same. This option demonstrates understanding of distribution but a lapse in understanding when to flip the inequality sign.

• Option D: $-12+20 x oun{ extgreater or equal to } -6 x+9$ This is our champion, guys! It correctly distributes $-4$ to get $-12+20x$ on the left side and it correctly maintains the original inequality sign $oun{ extgreater or equal to }$. This demonstrates a solid grasp of the distributive property and an understanding that this specific operation does not require flipping the inequality sign. It's the perfect, methodical first step towards solving the entire inequality. It’s all about precision in these early stages to ensure the final answer is accurate. Keep your eyes peeled for these common mistakes, and you'll navigate inequalities like a pro!

Moving Forward: The Next Steps in Solving

So, we've nailed the first step: distributing the $-4$ to get $-12+20 x oun{ extgreater or equal to } -6 x+9$. Now what? Solving inequalities is a journey, and this was just the beginning. Our goal is still to isolate the variable, $x$. The next logical moves involve gathering all the $x$ terms on one side and all the constant terms on the other. Remember all those rules we talked about? They're about to come into play. We want to be super careful, especially when we get to the point of needing to divide by a negative number. Let's keep that in mind as we proceed. The process is similar to solving equations, but with that crucial inequality rule always in the back of our minds. We'll add $6x$ to both sides to get all the $x$'s together: $-12 + 20x + 6x oun{ extgreater or equal to } -6x + 9 + 6x$. This simplifies to $-12 + 26x oun{ extgreater or equal to } 9$. Great! Now, we need to move the constant term, $-12$, to the right side. We do this by adding $12$ to both sides: $-12 + 26x + 12 oun{ extgreater or equal to } 9 + 12$. This gives us $26x oun{ extgreater or equal to } 21$. We're almost there! The final step to isolate $x$ is to divide both sides by $26$. Since $26$ is a positive number, we do not need to flip the inequality sign. So, $26x / 26 oun{ extgreater or equal to } 21 / 26$, which simplifies to $x oun{ extgreater or equal to } 21/26$. And there you have it! The complete solution set for our inequality. This journey from the initial messy form to the clear solution highlights the importance of each step, especially that critical first distribution. By understanding why each step is taken and which rules apply, you can confidently tackle even more complex algebraic challenges. Keep practicing, keep questioning, and don't be afraid to break down problems into smaller, manageable parts. You guys got this!