Unlocking Inequalities: A Step-by-Step Guide

Hey guys! Ever stumble upon an inequality and feel a little lost? Don't worry, we've all been there! Inequalities, just like equations, are mathematical statements, but instead of an equals sign (=), they use symbols like greater than (>), less than (<), greater than or equal to (≥), or less than or equal to (≤). Today, we're diving deep into how to solve an inequality, making sure you grasp every single step. We will tackle the inequality 2(4x - 3) ≥ -3(3x) + 5x. By the end of this guide, you will be able to solve this problem easily and confidently! Let's get started.

Understanding the Basics: Inequality Rules

Before we jump into the problem, let's brush up on a few key rules that will help us navigate inequalities like pros. The main goal when solving an inequality is pretty much the same as solving an equation: isolate the variable (in our case, 'x') on one side of the inequality sign. But there's a crucial difference you need to remember. This difference is super important! The difference is a single rule that can trip up even the best of us.

Here’s the golden rule: When you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign. That's right, you flip it! So, if you have a “greater than” sign (>) and you multiply or divide by a negative number, it becomes a “less than” sign (<), and vice versa. Keep this in mind, and you'll be golden. The other rules are the same as solving equations: you can add or subtract the same number from both sides, and you won’t change the solution. So, with these fundamentals in mind, let's get into the specifics of solving the given inequality.

Step-by-Step Solution: Solving the Inequality

Alright, let’s get down to business and solve the inequality 2(4x - 3) ≥ -3(3x) + 5x. We will solve this one step-by-step. Just follow along, and you'll become a master of inequalities in no time!

1. Distribute: First things first, we need to get rid of those parentheses. To do this, we'll distribute the numbers outside the parentheses to the terms inside. For the left side, we multiply 2 by both 4x and -3. On the right side, we multiply -3 by 3x. So, our inequality becomes: 8x - 6 ≥ -9x + 5x. See? Not so bad, right?

2. Combine Like Terms: Next, we'll combine the like terms on the right side of the inequality. We have -9x and 5x. Adding those together, we get -4x. Now our inequality looks like this: 8x - 6 ≥ -4x. We are slowly but surely getting there.

3. Isolate the Variable (x): Our goal is to get all the 'x' terms on one side and the constants on the other. Let's add 4x to both sides. This will cancel out the -4x on the right side. Our inequality becomes: 8x + 4x - 6 ≥ -4x + 4x. Simplifying, we get: 12x - 6 ≥ 0. Now, let's get rid of that -6. We add 6 to both sides. 12x - 6 + 6 ≥ 0 + 6. This simplifies to: 12x ≥ 6. We're getting closer! The x's are on the left, and the numbers are on the right.

4. Solve for x: Almost there! Now we need to isolate 'x'. To do this, we will divide both sides of the inequality by 12. 12x / 12 ≥ 6 / 12. Simplifying, we get: x ≥ 0.5. Ta-da! We’ve solved the inequality!

Understanding the Solution: What Does it Mean?

So, we've found that x ≥ 0.5. But what does that even mean? This solution tells us that any value of 'x' that is greater than or equal to 0.5 will make the original inequality true. Let's think about this for a sec. It means that 0.5 works, as do 1, 2, 3, 10, 100, and even 0.500001! If you pick any number greater than or equal to 0.5 and plug it into the original inequality, you'll find that the left side is always greater than or equal to the right side.

This also means that any number less than 0.5 will not work. For example, if you chose 0, then the original inequality is 2(4(0) - 3) ≥ -3(3(0)) + 5(0) which is -6 ≥ 0. This is obviously false. You can test a few values to convince yourself! The solution to an inequality is always a range of values, not just a single value like you get when you solve an equation. Understanding the solution allows you to see the scope of all valid solutions.

Choosing the Correct Answer: Let's Find the Right Option!

Now that we have solved the inequality and understand what the solution means, let's look at the multiple-choice options and select the correct one. The question provided us with the following options:

A. x ≥ 0.5 B. x ≥ 2 C. (-∞, 0.5] D. (-∞, 2]

We solved and found that x ≥ 0.5. This exactly matches the option A. Hence, Option A is the correct answer. The other options are incorrect. Option B indicates that 'x' should be greater than or equal to 2, which is more restrictive than our solution. Options C and D are a different way of representing the solution using interval notation. Interval notation is a way to represent a set of numbers using parentheses and brackets. In interval notation, a bracket [ ] indicates that the endpoint is included in the solution, and a parenthesis ( ) indicates that the endpoint is not included. So, in this context, C represents x ≤ 0.5 and D represents x ≤ 2. Since the solutions do not match our result, these are incorrect too. So, the correct answer is x ≥ 0.5.

Practice Makes Perfect: Try It Yourself!

Alright, guys! We've made it through! Now that you've seen how it's done, I strongly recommend practicing a few more inequalities on your own. Try solving different inequalities with different numbers and symbols. The more you practice, the more comfortable and confident you'll become. Remember to keep the rules in mind, especially the one about flipping the inequality sign when multiplying or dividing by a negative number. Keep practicing, and you'll be solving inequalities like a pro in no time! Keep it up, and you'll be inequality masters in no time!