Solving Inequalities: Find X When X/25 > 5

Hey guys! Let's dive into solving inequalities, specifically when we're faced with something like x/25 > 5. It might seem a bit daunting at first, but trust me, it's totally manageable. We'll break it down step by step, so you'll be a pro in no time. Think of inequalities as a way of saying that one thing is not equal to another, but rather greater than or less than it. In this case, we're looking for all the values of 'x' that make the left side of the inequality larger than 5. This involves using similar techniques to solving equations, but with a tiny twist that we'll cover. So, grab your pencils, and let's get started on unraveling this mathematical puzzle! We’re going to explore how to isolate 'x' and find the range of values that satisfy this inequality. By the end of this guide, you’ll not only know how to solve this specific problem but also have a solid understanding of the general principles involved in solving linear inequalities. We'll cover the critical steps, the common pitfalls to avoid, and some extra tips to ensure you ace these types of problems. Remember, practice makes perfect, so don't hesitate to try out similar problems on your own. Let's transform this seemingly complex inequality into something crystal clear and easy to handle. Are you ready? Let's jump right in and conquer this mathematical challenge together!

Understanding the Inequality

Okay, first things first, let’s really understand what the inequality x/25 > 5 is telling us. At its heart, this expression is a comparison. It's saying that when you take a certain number, which we're calling 'x', and divide it by 25, the result you get is bigger than 5. Think of it like having a pizza divided into 25 slices. The 'x' represents how many of those slices you need to have more than 5 whole pizzas. Makes sense, right? This sets the stage for our solution. We're not just looking for one single answer, but rather a range of values that 'x' can take to make this statement true. This is a key difference between solving equations and inequalities. Equations usually have a specific solution (or solutions), while inequalities often have a range of solutions. For instance, if x was 100, then x/25 would be 4, which is not greater than 5. So, 100 isn’t part of our solution set. But what if x was 150? Then x/25 would be 6, which is greater than 5. So, 150 would be a valid solution! Our job now is to find all such valid solutions in a systematic way. This is where the algebraic manipulation comes in, and it's surprisingly straightforward. We’ll use the properties of inequalities to isolate 'x' on one side and find the boundary that defines our solution set. Understanding the problem deeply is the first step towards solving it confidently. So let's move on and see how we can actually solve for 'x'.

Isolating 'x'

Now comes the fun part: actually solving for 'x'! Our main goal here is to get 'x' all by itself on one side of the inequality. To do this, we need to undo the operation that's currently affecting 'x'. In this case, 'x' is being divided by 25. So, what's the opposite of division? Multiplication! To isolate 'x', we're going to multiply both sides of the inequality by 25. This is a crucial step, and it's important to remember that whatever you do to one side of an inequality (or an equation), you must do to the other side to keep everything balanced. This maintains the truth of the inequality. So, we start with x/25 > 5. Multiply both sides by 25: (x/25) * 25 > 5 * 25 On the left side, the 25 in the numerator and the 25 in the denominator cancel each other out, leaving us with just 'x'. On the right side, 5 multiplied by 25 gives us 125. So, our inequality now looks like this: x > 125 And there you have it! We've successfully isolated 'x'. This new inequality tells us that 'x' must be any number greater than 125 to satisfy the original inequality. It's like we've found the magic boundary. Any number bigger than 125 will work, and any number 125 or smaller will not. This isolation process is the heart of solving inequalities (and equations), so make sure you understand each step. Next, we'll interpret what this solution means and how we can express it clearly.

Interpreting the Solution

Alright, we've arrived at the solution x > 125. But what does this really mean? It's not just a random collection of symbols; it's a powerful statement that tells us about the possible values of 'x'. In plain English, this inequality is saying that 'x' can be any number that is strictly greater than 125. Think about it this way: 125.000001 works, 126 works, 1000 works, and so on. The possibilities are endless, as long as the number is even slightly bigger than 125. This is the crucial difference between solving an equation and solving an inequality. If we had an equation like x = 125, there would be only one solution: 125. But here, we have a whole range of numbers that satisfy our inequality. Now, let's visualize this. Imagine a number line stretching out in both directions. The number 125 sits somewhere on that line. Our solution, x > 125, represents everything to the right of 125 on that line. It's a continuous set of numbers, zooming off towards infinity. We usually represent this on a number line with an open circle at 125 (to show that 125 itself is not included) and an arrow pointing to the right. This visual representation can be super helpful in understanding the nature of the solution. You can see at a glance that there are countless possibilities for 'x', all lined up neatly on the number line. Interpreting the solution isn't just about getting the right answer; it's about understanding what that answer truly represents in the context of the problem. Next, we'll talk about how to write down our solution in a clear and concise way.

Expressing the Solution

So, we know that x > 125, but how do we formally write this down as our final answer? There are a couple of standard ways to express solutions to inequalities, and we'll cover both. The first way is using inequality notation, which we've already been using. In this case, the solution is simply written as: x > 125 This is straightforward and clearly states that 'x' is greater than 125. It's the most direct way to express the solution, and it's perfectly acceptable in most situations. However, there's another way to express the solution, which is called interval notation. Interval notation uses parentheses and brackets to indicate the range of values that 'x' can take. Parentheses ( ) are used to show that an endpoint is not included in the solution, while brackets [ ] are used to show that an endpoint is included. Since our solution is x > 125, we want to include all numbers greater than 125, but not 125 itself. So, we use a parenthesis for 125. On the other end, the solution goes on forever towards positive infinity. We represent infinity with the ∞ symbol, and we always use a parenthesis with infinity because infinity is not a specific number that can be included. Therefore, in interval notation, our solution is written as: (125, ∞) This notation means