Inequality Solution: 4x - 7 >= 5x + 2

Hey guys! Today we're diving into a super common but sometimes tricky topic in math: solving inequalities. We've got a specific one to tackle: $4 x-7 ">=" 5 x+2$. Don't let the 'greater than or equal to' symbol throw you off; the process is really similar to solving regular equations. We're aiming to isolate that variable, 'x', and figure out what values of 'x' make this statement true. We'll break it down step-by-step, making sure you guys can follow along and get a solid grasp on how to find the solution. Plus, we'll look at the multiple-choice options provided – A, B, C, and D – to see which one correctly represents the solution set for this inequality. So grab your notebooks, maybe a snack, and let's get this math party started!

Understanding Inequalities: The Basics

Alright, let's kick things off by making sure we're all on the same page about what inequalities are and how they differ from equations. Think of an equation like a perfectly balanced scale: $4 x-7 = 5 x+2$. Whatever you do to one side, you must do to the other to keep it balanced. An inequality, like our problem $4 x-7 ">=" 5 x+2$, is more like a scale that can be tipped. The 'greater than or equal to' symbol ($ ">=" $**) means that the left side is either bigger than the right side, or it's exactly the same. Our goal here is still to find the values of 'x' that satisfy this condition. The key difference we need to remember, which is super important, is what happens when we multiply or divide both sides of an inequality by a *negative* number. When that happens, the inequality sign flips! If it was 'greater than' (**$ > $**), it becomes 'less than' (**$ < $**), and vice versa. For 'greater than or equal to' (**$ ">=" $**) and 'less than or equal to' (**$ "<=" $**), they flip to their opposite too. This is a crucial rule, so keep it in the back of your minds as we work through our problem. For our specific inequality, **$$4 x-7 ">=" 5 x+2$$, we'll be moving terms around, and we need to be mindful of that sign-flipping rule, although in this particular case, we might not even need to divide by a negative, which is always a bonus!

Step-by-Step Solution of the Inequality

Now for the main event, guys! Let's solve $4 x-7 ">=" 5 x+2$ together. Our primary mission is to get all the 'x' terms on one side of the inequality and all the constant numbers on the other. It doesn't matter which side you choose for the 'x's, but sometimes picking the side that results in a positive coefficient for 'x' can make things a little easier. Let's try moving the 'x' terms to the right side this time. To do that, we'll subtract $4x$ from both sides:

$$4 x - 7 - 4x ">=" 5 x + 2 - 4x$$

This simplifies to:

$$-7 ">=" x + 2$$

See? We successfully got all the 'x' terms to one side. Now, we need to isolate the 'x' by moving the constant term, $+2$, to the left side. We do this by subtracting $2$ from both sides:

$$-7 - 2 ">=" x + 2 - 2$$

This leaves us with:

$$-9 ">=" x$$

Now, we have the 'x' on the right side and the number on the left. It's perfectly correct as it is, but most people find it easier to read when the variable is on the left. So, we can rewrite $-9 ">=" x$ by flipping both the variable and the number, and we must also flip the inequality sign. So, $-9 ">=" x$ becomes $x "<=" -9$.

Think about it: if -9 is greater than or equal to x, that means x must be less than or equal to -9. For example, if x was -10, then -9 is indeed greater than -10. If x was -9, then -9 is equal to -9. But if x was -8, then -9 is not greater than or equal to -8. So, our solution set is all numbers less than or equal to -9. This is a really common place for folks to get tripped up, so always double-check that sign flip when you're rewriting!

Comparing with the Options

So, we've worked through the inequality $4 x-7 ">=" 5 x+2$ and arrived at our solution: $x "<=" -9$. Now, let's look at the multiple-choice options provided to see which one matches our hard work:

A. $x ">=" -9$ B. $x "<=" -9$ C. $x ">=" 9$ D. $x "<=" 9$

By comparing our result, $x "<=" -9$, with these options, it's clear that Option B is the correct answer. It perfectly matches the solution we derived. It's always a good idea to plug a value back into the original inequality just to be absolutely sure, especially if you're feeling a bit uncertain. Let's test x = -10 (which is less than or equal to -9).

Original inequality: $4x - 7 ">=" 5x + 2$ Substitute x = -10:

$$4(-10) - 7 ">=" 5(-10) + 2$$

$$-40 - 7 ">=" -50 + 2$$

$$-47 ">=" -48$$

Is -47 greater than or equal to -48? Yes, it is! This confirms our solution is correct. Now, let's try a value that is not in our solution set, say x = 0 (which is greater than -9).

Substitute x = 0:

$$4(0) - 7 ">=" 5(0) + 2$$

$$0 - 7 ">=" 0 + 2$$

$$-7 ">=" 2$$

Is -7 greater than or equal to 2? No, it's definitely not. This further solidifies that our solution $x "<=" -9$ is indeed the correct one.

Final Thoughts on Inequalities

And there you have it, guys! We've successfully solved the inequality $4 x-7 ">=" 5 x+2$ and found that the solution is $x "<=" -9$, which corresponds to Option B. The key takeaways here are to treat inequalities much like equations when you're isolating the variable, but always, always remember to flip the inequality sign if you multiply or divide by a negative number. It's a small detail that can make a huge difference in getting the right answer. Practicing these kinds of problems will make you more comfortable and confident. Don't be afraid to go back and review the steps, especially the part where we rewrote the inequality to have 'x' on the left. Understanding why the sign flips is crucial for mastering inequalities. Keep practicing, keep questioning, and you'll nail these in no time. If you guys have any other tricky inequalities you want to break down, drop them in the comments! Happy solving!