Solve & Graph 9x - 2 ≤ 5x + 10: Your Easy Guide

Introduction: Demystifying Inequalities

Hey there, math explorers! Ever looked at a problem with a funky symbol instead of a regular equals sign and thought, "What the heck is that?" Well, guys, you've probably stumbled upon an inequality. Today, we're going to dive headfirst into the world of inequalities, specifically learning how to solve and graph the inequality 9x - 2 ≤ 5x + 10. Don't sweat it; it's way less intimidating than it looks, and by the end of this article, you'll be tackling these problems like a total pro. Inequalities are super cool because they don't just tell us one specific answer; they show us a whole range of possible solutions. Think about it: when you're told the speed limit is 60 MPH, that means you can drive anything less than or equal to 60 MPH, not just exactly 60. That's an inequality in action! Learning to solve and graph inequalities like our example, 9x - 2 ≤ 5x + 10, is a fundamental skill in algebra that opens doors to understanding more complex mathematical concepts and even real-world scenarios. We're talking about understanding limitations, boundaries, and possibilities in a numerical sense. This guide is designed to break down every single step, making it super clear and easy to follow, even if you feel like math isn't exactly your jam. We'll walk through isolating the variable, understanding what the solution means, and then visually representing it on a number line. Our goal is not just to give you the answer, but to empower you with the understanding of why each step is taken. So, grab a pen and paper, maybe a snack, and let's get ready to conquer 9x - 2 ≤ 5x + 10 together. This isn't just about memorizing steps; it's about building a solid foundation in mathematical reasoning, making you more confident and capable when faced with any algebraic challenge. We'll make sure to hit all the key points, emphasize important rules, and share some helpful tips along the way, ensuring you get maximum value from this learning journey.

Step-by-Step Breakdown: Solving 9x - 2 ≤ 5x + 10

Alright, folks, let's roll up our sleeves and get down to business: solving the inequality 9x - 2 ≤ 5x + 10. Just like solving regular equations, our main goal here is to get 'x' all by itself on one side of the inequality symbol. The process is remarkably similar to solving equations, but there's one crucial rule we need to remember when dealing with inequalities, which we'll highlight when it comes up. For now, let's treat the '≤' symbol almost like an equals sign and start moving terms around. We'll tackle this in a few easy-to-digest steps. First, we'll gather all the 'x' terms together, then we'll deal with the constant numbers, and finally, we'll isolate 'x' completely. This systematic approach will ensure we don't miss anything and arrive at the correct solution for 9x - 2 ≤ 5x + 10. It's all about being organized and taking it one piece at a time. Remember, every step has a purpose, moving us closer to unraveling the mystery of what 'x' truly represents in this specific mathematical relationship. This methodical approach is key to understanding and mastering not just this specific inequality, but all future algebraic challenges you might encounter. We want to build your confidence and make you feel like you've got this, every single time.

Isolate the Variable: Gathering Your 'X's

Our first mission in solving 9x - 2 ≤ 5x + 10 is to get all the 'x' terms on one side of the inequality. It doesn't really matter which side you choose, but a common practice is to move the 'x' term with the smaller coefficient to the side with the larger coefficient, just to keep things positive and a little easier to manage. In our case, we have 9x on the left and 5x on the right. Since 5x is smaller than 9x, let's move 5x from the right side to the left side. To do this, we perform the opposite operation. Since 5x is being added (it's positive), we'll subtract 5x from both sides of the inequality. Remember, whatever you do to one side, you must do to the other side to keep the inequality balanced! This is a fundamental rule in algebra, whether you're dealing with equations or inequalities. So, let's write it out:

9x - 2 ≤ 5x + 10 9x - 5x - 2 ≤ 5x - 5x + 10 4x - 2 ≤ 0 + 10 4x - 2 ≤ 10

See that? We've successfully gathered our 'x' terms, and now we have 4x on the left side. The inequality sign, ≤, remains exactly the same because we only performed subtraction. It's crucial to understand that subtracting a value from both sides does not change the direction of the inequality symbol. This is one of those rules that sometimes trips people up, but it's pretty straightforward. We've simplified the expression considerably, and we're one big step closer to isolating 'x'. This initial step is often the first hurdle for many, but by breaking it down, you can see it's just basic arithmetic applied consistently. The key is balance; ensuring that the scale remains tipped in the same direction, or remains level, no matter what operations you apply, as long as you apply them universally. Keep that positive attitude, because we're doing great so far in solving 9x - 2 ≤ 5x + 10!

Clear the Constants: Moving the Numbers Around

Fantastic work on the first step, everyone! Now that all our 'x' terms are cozy on the left side, the next part of solving 9x - 2 ≤ 5x + 10 is to get rid of any constant numbers hanging out with 'x' on that side. In our current situation, we have 4x - 2 ≤ 10. The constant number on the left side is -2. Just like before, to move this -2 to the other side of the inequality, we need to perform the opposite operation. Since we are subtracting 2 from 4x, we will add 2 to both sides of the inequality. Again, balance is key here! This step effectively