Solving Inequalities: A Step-by-Step Guide

Hey everyone! Today, we're diving into the world of inequalities, specifically tackling the question: Which is the correct first step in solving $5 - 2x < 8x - 3$? This might seem like a small problem, but understanding the fundamentals of solving inequalities is super important in math. We'll break down the steps, explore the answer choices, and make sure you've got a solid grasp of the concepts. So, grab your pencils, and let's get started!

Understanding the Basics of Inequalities

Alright, before we jump into the problem, let's quickly recap what inequalities are all about. Inequalities, guys, are mathematical statements that compare two expressions using symbols like < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). Unlike equations, which use an equals sign (=), inequalities show a range of possible values. The goal when solving an inequality is to find the values of the variable (in our case, x) that make the inequality true. The steps for solving inequalities are pretty similar to solving equations, but there's a crucial rule to remember: When you multiply or divide both sides of an inequality by a negative number, you must flip the direction of the inequality sign. Keep this in mind, and you'll be golden. Understanding these basics is critical for answering our question correctly. Now, let's see how we can apply these concepts to our specific problem.

So, our initial inequality is $5 - 2x < 8x - 3$. We want to isolate the variable x on one side of the inequality. The best first step usually involves either combining the x terms or moving the constant terms. We need to look at our answer choices and see which one follows the rules of algebra. Remember, we want to perform operations that maintain the balance of the inequality. That means whatever we do to one side, we have to do to the other. Are you ready to dive into the answer choices? Let's go! I know you can do it!

Analyzing the Answer Choices

Okay, guys, let's dissect each answer choice to figure out the correct first step in solving the inequality $5 - 2x < 8x - 3$. This is where the real fun begins! Each answer choice presents a different manipulation of the original inequality. We'll evaluate each one to see if it follows the rules of algebra and, more importantly, whether it's a valid first step towards isolating x. Remember, we're aiming to get x by itself on one side of the inequality. We need to choose the option that logically and correctly moves us closer to that goal. Let's see what we've got:

• A. $3x < 8x - 3$: This choice suggests that the 5 has been removed somehow. This is incorrect. There's no valid algebraic operation that directly transforms the original inequality into this form in a single, correct step. This is a red flag! When we are solving the inequality we must keep in mind to do the same operation in both sides, that way we are sure the inequality holds its value.
• B. $5 < 6x - 3$: This looks like it was created by adding $2x$ to both sides, so $-2x$ on the left side becomes zero and $8x$ on the right side becomes $10x$. However, this is wrong, because $8x + 2x = 10x$. So this can be discarded as well.
• C. $2 - 2x < 8x$: This one suggests that 5 was subtracted from the left side and 3 was removed from the right side. This option is not correct because subtracting the 5 in the left side won't make it become a 2. Also, the 3 was not subtracted from the right side. This looks very wrong, you can remove this one.
• D. $5 < 10x - 3$: This is the most promising option! This looks like we've added $2x$ to both sides of the original inequality. Let's check: Starting with $5 - 2x < 8x - 3$, adding $2x$ to both sides gives us $5 < 10x - 3$. This is a valid algebraic operation and it's a step in the right direction to isolate x. So this one seems correct. Keep in mind that when we solve inequalities or equations, the goal is always to isolate the variable, by doing the inverse operation.

By carefully examining each option and applying our knowledge of algebraic rules, we can confidently identify the correct first step. But hey, it’s not just about getting the right answer; it's about understanding why that answer is correct. This is how we build a strong foundation in math, guys. Now, let’s go over the correct solution.

The Correct First Step and Why

So, after careful consideration, the correct first step in solving $5 - 2x < 8x - 3$ is D. $5 < 10x - 3$. This is because, as we analyzed before, this step is achieved by adding $2x$ to both sides of the original inequality. Here's how it works:

1. Original Inequality: $5 - 2x < 8x - 3$
2. Add $2x$ to both sides: $(5 - 2x) + 2x < (8x - 3) + 2x$
3. Simplify: $5 < 10x - 3$

This step is valid because we've performed the same operation (adding $2x$) on both sides of the inequality, thereby maintaining its balance. The next step would be to add 3 to both sides to get the constants together, and then divide by 10 to isolate x. This move gets us closer to our goal of isolating x and solving the inequality. Remember that there are multiple ways to solve an inequality, but you can only do operations following the mathematical rules, if not you will get wrong answers. The correct answer leads us toward isolating x! Awesome, right?

Continuing to Solve the Inequality

Now that we've found the correct first step, let's quickly walk through the rest of the solution to give you a complete picture. This will solidify your understanding and show you how the process unfolds. After getting to $5 < 10x - 3$, our next move would be to isolate the x. Here's how it goes:

1. Add 3 to both sides: $5 + 3 < 10x - 3 + 3$, which simplifies to $8 < 10x$.
2. Divide both sides by 10: $\frac{8}{10} < \frac{10x}{10}$, which simplifies to $0.8 < x$ or $x > 0.8$.

And there you have it! The solution to the inequality is $x > 0.8$. This means that any value of x greater than 0.8 will satisfy the original inequality. See? It's all about breaking down the problem step-by-step. Now you know how to solve this kind of inequalities. Keep practicing, and you'll become a pro in no time! Remember that you must master basic rules, so you don't get stuck in the middle of a problem. Practice makes perfect, and with each inequality you solve, you'll build your confidence and your skills. Now, let's summarize all we've learned today.

Summary and Key Takeaways

Alright, let's wrap things up with a quick recap of what we've covered today. We started with the question: Which is a correct first step in solving $5 - 2x < 8x - 3$? We analyzed the problem, looked at the rules of inequalities, and worked through the answer choices. The correct first step is $5 < 10x - 3$, which we arrived at by adding $2x$ to both sides. We then finished solving the inequality, demonstrating the complete process. Here's a summary of the key takeaways:

• Understanding Inequalities: Inequalities compare two expressions using symbols like <, >, ≤, and ≥. The goal is to find the values of the variable that make the inequality true.
• Key Rule: When multiplying or dividing both sides of an inequality by a negative number, flip the inequality sign.
• Correct First Step: In the given problem, the correct first step is $5 < 10x - 3$, achieved by adding $2x$ to both sides.
• Solving Process: After the first step, continue isolating the variable using valid algebraic operations (add/subtract, multiply/divide).

I hope this guide has helped you understand the process of solving inequalities. Keep practicing and remember the key rules. You've got this! And guys, always remember: math is a journey, not a destination. Keep learning, keep exploring, and don't be afraid to ask for help when you need it. Until next time, keep solving, keep exploring, and keep the math vibes strong!