Solving Inequalities The Value Of X In 3(x-4) ≥ 5x + 2

Inequalities can sometimes feel like puzzles, but don't worry, guys! We're going to break down this problem step by step and find the value of $x$ that fits the solution. The inequality we're tackling today is 3(x-4) ackslash geq 5x + 2. This might look a little intimidating at first, but with a bit of algebraic magic, we'll solve it together. Remember, the key to solving inequalities, just like equations, is to isolate the variable – in this case, $x$. We need to get $x$ all by itself on one side of the inequality. So, let's roll up our sleeves and get started!

Step 1: Distribute the 3

The first thing we need to do is get rid of those parentheses. We'll do this by distributing the 3 on the left side of the inequality. That means we multiply both the $x$ and the -4 inside the parentheses by 3. This gives us:

3 * x - 3 * 4 ackslash geq 5x + 2

Which simplifies to:

3x - 12 ackslash geq 5x + 2

Now our inequality looks a little cleaner and easier to work with. We've successfully eliminated the parentheses, which is a crucial first step in solving for $x$. Distributing correctly is super important because it ensures we're maintaining the balance of the inequality. Think of it like this: we're making sure both sides of the scale stay in proportion. If we mess up the distribution, the whole solution could be thrown off. So, always double-check your distribution to make sure you've multiplied correctly. This simple step sets the foundation for the rest of the solution, so let's move on to the next part!

Step 2: Move the x Terms to One Side

Okay, now we need to get all the $x$ terms on one side of the inequality. It doesn't matter which side we choose, but let's go for the side that will keep our $x$ term positive. In this case, we have $3x$ on the left and $5x$ on the right. To keep things positive, let's move the $3x$ to the right side. We can do this by subtracting $3x$ from both sides of the inequality. Remember, whatever we do to one side, we have to do to the other to keep everything balanced.

3x - 12 - 3x ackslash geq 5x + 2 - 3x

This simplifies to:

-12 ackslash geq 2x + 2

See? We're making progress! The $x$ term is now only on the right side, and we're one step closer to isolating it. Moving the $x$ terms to one side is a key strategy in solving inequalities. It's like gathering all the ingredients you need for a recipe in one place before you start cooking. By grouping the $x$ terms together, we can then focus on isolating $x$ and figuring out its value. It's all about organization and making the problem more manageable. So, let's keep going – we're almost there!

Step 3: Move the Constant Terms to the Other Side

Next up, we need to isolate the $x$ term further by moving the constant terms (the numbers without $x$) to the other side of the inequality. We have a +2 on the right side with the $2x$, so let's get rid of it by subtracting 2 from both sides. Again, keeping that balance is crucial!

-12 - 2 ackslash geq 2x + 2 - 2

This simplifies to:

-14 ackslash geq 2x

Awesome! Now we have all the $x$ terms on one side and all the constant terms on the other. We're really honing in on the solution here. Moving the constant terms is like separating the dry ingredients from the wet ingredients in a baking recipe – you need to deal with them separately before you can combine them properly. By isolating the $x$ term, we're setting ourselves up for the final step, which is to get $x$ completely alone. So, let's finish strong and get that value of $x$!

Step 4: Isolate x by Dividing

We're almost there! The final step is to get $x$ completely by itself. Right now, we have $2x$ on the right side, which means $x$ is being multiplied by 2. To undo this multiplication, we need to divide both sides of the inequality by 2.

-14 / 2 ackslash geq 2x / 2

This simplifies to:

-7 ackslash geq x

Or, we can rewrite this as:

x ackslash leq -7

But hold on a second! Remember that when we divide (or multiply) an inequality by a negative number, we need to flip the inequality sign. However, in this case, we divided by a positive 2, so we don't need to flip the sign. Phew!

Dividing to isolate $x$ is the final piece of the puzzle. It's like finding the missing ingredient that completes the dish. By dividing both sides by the coefficient of $x$, we're essentially scaling down the inequality to find the exact range of values that $x$ can take. This step requires careful attention, especially when dealing with negative numbers, as we need to remember that crucial rule about flipping the inequality sign. But in this case, we're good to go! We've successfully isolated $x$ and found the solution.

Step 5: Check the Answer Choices

Okay, so we've found that x ackslash leq -7. This means that $x$ can be any number less than or equal to -7. Now, let's look at our answer choices and see which one fits this condition:

• -10
• -5
• 5
• 10

Which of these numbers is less than or equal to -7? That's right, it's -10!

Checking the answer choices against our solution is like taste-testing your dish to make sure it's just right. We've done all the hard work of solving the inequality, and now we need to make sure our answer makes sense in the context of the problem. By plugging in the answer choices, we can quickly verify which one satisfies the inequality and gives us the correct solution. It's a simple but crucial step that ensures we're confident in our final answer. So, let's celebrate – we've found the value of $x$!

Final Answer

The value of $x$ in the solution set of 3(x-4) ackslash geq 5x + 2 is -10.

Conclusion:

So, there you have it! We've successfully solved the inequality and found the value of $x$. Remember, guys, the key to tackling these problems is to break them down into smaller, manageable steps. Distribute, move the $x$ terms, move the constants, isolate $x$, and always double-check your work! With a little practice, you'll be inequality-solving pros in no time. Keep up the great work, and happy problem-solving!