Solving Double Inequalities: Find Integer Solutions

Hey guys! Today, we're diving into the world of double inequalities and figuring out how to solve them. It might sound a bit intimidating, but trust me, it's totally manageable. We'll break down a specific example step-by-step and also talk about how to find integer solutions. So, if you've ever scratched your head at these problems, you're in the right place! Let's get started and make math a little less scary, one inequality at a time.

Understanding Double Inequalities

When you're faced with a double inequality, like the one we're tackling today, understanding the core concepts is super important. A double inequality is basically a combination of two inequalities slapped together, and they're usually connected by the variable sitting smack-dab in the middle. Think of it like a math sandwich where the variable is the delicious filling. To solve it, you've got to work on all parts of the inequality at the same time – what you do to one side, you've got to do to all the sides. It’s like making sure every slice of bread gets the same amount of butter. This ensures that you're keeping the whole thing balanced and that your final answer is spot-on. The goal here is to isolate the variable, just like in a regular equation or inequality, but you're doing it across three sections instead of two. It's a bit like juggling, but once you get the hang of it, you'll be solving these things like a pro. So, let's jump into the nitty-gritty and see how this works in practice.

Breaking Down the Inequality: -5 < 9 - 2x ≤ 4

Okay, let's dive into breaking down this specific inequality: -5 < 9 - 2x ≤ 4. This might look a bit like a jumble of numbers and symbols right now, but we're going to untangle it piece by piece. The first thing to recognize is that this isn't just one inequality; it’s two in one! We've got -5 is less than 9 minus 2x, and at the same time, 9 minus 2x is less than or equal to 4. See? Two inequalities hanging out together. The key to cracking this nut is to remember that we need to isolate 'x', but whatever operation we perform, we have to do it on all three parts: the left side (-5), the middle (9 - 2x), and the right side (4). Think of it as a mathematical group hug – everyone gets the same squeeze! This keeps the whole inequality balanced and ensures we're on the right track to finding our solution. So, with that in mind, let’s start peeling back the layers and get 'x' all by itself. We'll take it one step at a time, and before you know it, we'll have this inequality solved. Ready? Let’s go!

Step 1: Isolating the Term with 'x'

Alright, let’s get to isolating the term with 'x' in our double inequality: -5 < 9 - 2x ≤ 4. The first thing we want to do is get that '-2x' term all by its lonesome in the middle. Right now, it's hanging out with a '+9', and we need to get rid of it. How do we do that? We use the magic of inverse operations! Since we have a '+9', we're going to subtract 9. But remember the golden rule of double inequalities: what we do to one part, we do to all parts. So, we subtract 9 from the left side, the middle, and the right side. This gives us: -5 - 9 < 9 - 2x - 9 ≤ 4 - 9. If we simplify that, we get: -14 < -2x ≤ -5. See? We're making progress! The '-2x' is getting closer to being on its own. Now, we've got a slightly simpler inequality to deal with. We’re one step closer to uncovering the value (or values) of 'x'. Stick with me, and we'll have this cracked in no time.

Step 2: Solving for 'x'

Now comes the crucial part: solving for 'x'. We've got -14 < -2x ≤ -5, and the mission is to get 'x' all by itself. Currently, 'x' is being multiplied by -2, so what's the opposite of multiplication? Division! We're going to divide all parts of the inequality by -2. But hold on a second! Here’s a super important rule to remember: when you multiply or divide an inequality by a negative number, you've got to flip the direction of the inequality signs. It's like the inequality is doing a little somersault. So, when we divide by -2, our '<' becomes a '>' and our '≤' becomes a '≥'. Let's do it: -14 / -2 > -2x / -2 ≥ -5 / -2. Simplifying that gives us: 7 > x ≥ 2.5. Awesome! We've got 'x' by itself. But let's rewrite this so it makes a bit more sense in terms of how we usually read inequalities, with the smaller number on the left. So, we flip the whole thing around to get: 2.5 ≤ x < 7. There you have it! We've solved the inequality. 'x' is greater than or equal to 2.5 and less than 7. Feels good, right? But we're not done yet; we still need to find those integer solutions.

Finding Integer Solutions

Now that we've conquered the inequality itself, let's zoom in on finding the integer solutions. Remember, integers are whole numbers – no fractions or decimals allowed! Our solution to the inequality is 2.5 ≤ x < 7. This means 'x' can be any number between 2.5 and 7, including 2.5 but not including 7 (because of the 'less than' sign). So, which whole numbers fit into this range? Let’s think it through. We know 2.5 is our starting point, so the first integer that 'x' can be is 3. Then we've got 4, 5, and 6. But 7 is a no-go because 'x' has to be strictly less than 7. So, our integer solutions are 3, 4, 5, and 6. That's it! We've found all the whole number answers that make our original double inequality true. This step is super important because sometimes the question isn’t just about solving the inequality, it's about pinpointing the specific types of numbers that fit the solution. And in this case, those numbers are integers. So, give yourself a pat on the back – you're becoming a pro at this!

Listing the Integer Solutions

Let's get super clear and list the integer solutions we found. After solving the double inequality 2.5 ≤ x < 7, we figured out that 'x' can be any number between 2.5 and 7, but not including 7 itself. Now, we're only interested in the whole numbers – the integers – that fit this range. So, let's line them up: The integers that are greater than or equal to 2.5 are 3, 4, 5, and 6. Boom! That's our list. We don't include 2 because it's less than 2.5, and we don't include 7 because 'x' has to be strictly less than 7. This list is the final answer to the second part of our problem – finding the integer solutions. It’s like the cherry on top of our inequality-solving sundae. We've not only solved for 'x', but we've also identified the specific whole numbers that make the inequality true. Nice work!

Counting the Integer Solutions

Okay, we've got our integer solutions all lined up: 3, 4, 5, and 6. Now, the final piece of the puzzle is counting these integer solutions. This is actually the easiest part – we just need to count how many numbers are on our list. Let's see... we've got 3, that's one, then 4, that's two, then 5, making three, and finally 6, bringing us to a grand total of four integer solutions. That's it! The number of integer solutions to our double inequality -5 < 9 - 2x ≤ 4 is four. Sometimes, in math problems, it's not enough just to find the solutions; you need to quantify them – give a specific number. And that's exactly what we've done here. We've gone from solving the inequality to pinpointing the integer solutions and then counting how many there are. We’ve nailed it! You’re doing great, guys.

Conclusion

We've reached the end of our double inequality adventure, and guess what? We totally crushed it! We started with the inequality -5 < 9 - 2x ≤ 4, and we walked through each step, like true math detectives. First, we talked about what double inequalities are and how to approach them. Then, we got our hands dirty and isolated the term with 'x', making sure to do the same thing to all parts of the inequality. Remember subtracting 9 from all sides? Good job! Next, we solved for 'x', and here's where we had to remember that sneaky rule about flipping the inequality signs when dividing by a negative number. We ended up with 2.5 ≤ x < 7, which is a fantastic solution. But we didn't stop there! We went on to find the integer solutions, those whole numbers that fit our inequality. We listed them out: 3, 4, 5, and 6. And finally, we counted them up and found that there are four integer solutions in total. That’s a full investigation, from start to finish! You’ve not only learned how to solve a double inequality but also how to pinpoint and count specific types of solutions. You're building some serious math skills here, and that's something to be proud of. Keep up the great work, and remember, every math problem is just another puzzle waiting to be solved!