Solve 5(x+5)<85: Your Easy Inequality Guide
Hey there, math explorers! Ever stared at an inequality like 5(x+5) < 85 and wondered, "What in the world am I supposed to do with this?" You're definitely not alone, guys! Many people find inequalities a bit trickier than regular equations, but trust me, by the end of this super friendly guide, you'll be tackling them like a pro. We're not just going to solve this specific problem, we're going to break down the whole concept of inequalities, understand why certain steps are taken, and make sure you feel totally confident in finding that elusive solution set. Our main goal today is to unravel the mystery behind expressions like 5(x+5) < 85 and figure out what values of 'x' actually make this statement true. It might look a little intimidating at first glance with that 'less than' sign and the parentheses, but it's really just a puzzle waiting to be solved. We'll go through it step-by-step, ensuring every little detail is clear. Think of it like a fun mathematical adventure where we uncover the hidden range of numbers that satisfy this condition. We’ll dive into what inequalities actually mean compared to equations, how to handle the distribution, and most importantly, how to correctly interpret the final answer. So, buckle up, grab a virtual coffee, and let's conquer this inequality together, making mathematics not just understandable, but genuinely enjoyable!
What Are Inequalities, Really?
Alright, so before we jump into 5(x+5) < 85, let's chat a bit about what inequalities actually are. In the simplest terms, an inequality is a mathematical statement that shows the relationship between two expressions, indicating that one is not equal to the other. Instead, it might be greater than, less than, greater than or equal to, or less than or equal to. Think about it: an equation uses an equals sign (=) to say, "Hey, these two sides are exactly the same." For example, x + 2 = 5 means x must be 3. No other number will work! But an inequality? It's way more chill. It says, "This side is bigger than that side," or "This side is smaller than or equal to that side." It's like saying, "You need to drive less than 60 mph on this road," rather than "You must drive 60 mph." There's a whole range of possibilities, not just one single answer. The common symbols you'll see are: < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). Understanding these symbols is super important because they totally change the solution set – which is just a fancy way of saying "all the numbers that make this statement true." For our problem, 5(x+5) < 85, we're looking for all the 'x' values that make the expression on the left strictly less than 85. This means 85 itself isn't included, and we're exploring an entire spectrum of numbers. Imagine it on a number line: instead of pointing to just one spot, we'll be highlighting a whole section! This fundamental difference between equations and inequalities is key to solving them correctly and interpreting their results. It's about defining a boundary or a limit, rather than a fixed point. So, when you see that inequality sign, get ready to think about a whole bunch of numbers, not just one! We're talking about a world of possibilities that satisfy the condition, which makes them incredibly useful for modeling real-world situations like speed limits, budget constraints, or even minimum requirements for passing a test. It's truly a foundational concept in algebra that helps us describe relationships where exact equality isn't always the case, giving us a powerful tool to analyze a broader spectrum of scenarios.
Diving Deep into 5(x+5)<85: Step-by-Step Breakdown
Okay, guys, it's time to roll up our sleeves and get down to business with our specific problem: 5(x+5) < 85. Don't let the parentheses scare you; we're going to break it down into super manageable steps, just like we'd solve a regular equation. The process is remarkably similar, with just one crucial rule to remember about inequalities that we'll highlight when we get there. Our aim is to isolate 'x' on one side of the inequality sign, which will reveal the range of values that 'x' can take. This step-by-step approach ensures that we don't miss anything and that our final answer is absolutely correct. So, let's get cracking and unveil the solution to this mathematical mystery! We're building confidence one step at a time, making sure that each phase of the solution makes perfect sense before moving on to the next. This methodical approach is your best friend when dealing with any algebraic expression, especially inequalities where a small mistake can lead to a completely different solution set. We're going to treat this like a quest, transforming the complex into the clear, and turning a potentially daunting problem into a triumph of understanding.
Step 1: Distribute and Conquer
The very first thing you'll want to do when you see parentheses in an expression like 5(x+5) < 85 is to get rid of them using the distributive property. Remember that awesome rule from earlier math classes? It basically says that the number outside the parentheses multiplies every term inside the parentheses. So, here, the '5' needs to multiply both 'x' and '5'. Let's do it: 5 multiplied by 'x' gives us 5x. And 5 multiplied by '5' gives us 25. So, our inequality now transforms from 5(x+5) < 85 into a much friendlier looking 5x + 25 < 85. See? Not so bad, right? We've successfully expanded the expression and removed those tricky parentheses, paving the way for us to further isolate 'x'. This is a critical first move in many algebraic problems, simplifying the expression and making subsequent steps much clearer. Always make sure to distribute correctly, multiplying the outside term by each and every term inside the parentheses. A common mistake here is to only multiply the 'x' and forget the '5', which would throw off the entire calculation. Double-check your distribution, and you'll be off to a fantastic start in solving this inequality. This step is about setting the stage, transforming the initial presentation into a more workable form, just like untangling a knot before you can properly tie it. It simplifies the complexity and makes the next actions intuitive and straightforward.
Step 2: Isolate the Variable – Moving the Constants
Now that we have 5x + 25 < 85, our next mission is to start isolating 'x'. To do this, we need to get rid of that '+25' that's hanging out with our '5x'. How do we do that? By doing the opposite operation! Since 25 is being added, we'll subtract 25 from both sides of the inequality. This is super important: whatever you do to one side, you must do to the other side to keep the inequality balanced, just like a seesaw. So, we'll subtract 25 from 5x + 25 and subtract 25 from 85. On the left side, +25 - 25 cancels out, leaving us with just 5x. On the right side, 85 - 25 gives us 60. So, our inequality now looks like this: 5x < 60. We're getting super close to finding 'x' on its own! This step is all about moving the constant terms away from the variable term. It’s a fundamental principle in algebra – maintaining balance. If you don't perform the same operation on both sides, you're essentially changing the problem, and your solution will be incorrect. Always remember to use inverse operations; addition undoes subtraction, and subtraction undoes addition. This systematic approach ensures the integrity of your inequality as you work towards isolating 'x'. Think of it as peeling layers off an onion, one careful step at a time, until you reveal the core. The goal is to simplify the expression, bringing us closer to a clear understanding of what 'x' truly represents in this specific scenario. It's crucial to be meticulous here, as a simple arithmetic error in this step can propagate through the rest of your calculations.
Step 3: Final Touches – Getting 'x' All Alone
We're in the home stretch, folks! Our inequality is currently 5x < 60. To finally get 'x' completely by itself, we need to deal with that '5' that's multiplying it. Again, we do the opposite operation. Since 'x' is being multiplied by 5, we'll divide both sides of the inequality by 5. On the left side, 5x divided by 5 simply leaves us with x. On the right side, 60 divided by 5 gives us 12. So, our final, glorious inequality is: x < 12. And here's that super important rule I mentioned earlier, so listen up! When you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign. For example, if it was '<', it would become '>', and vice-versa. However, in our case, we divided by a positive 5, so the sign stays exactly the same. Phew! No flipping needed here. This is perhaps the most common mistake students make with inequalities, so always, always double-check if the number you're multiplying or dividing by is positive or negative. Since 5 is positive, our '<' sign remains '<'. This distinction is absolutely critical for obtaining the correct solution set. Failing to flip the sign when dividing by a negative number would lead you to an entirely wrong set of values for 'x', which means your solution would be incorrect. Always pause and consider the sign of the divisor or multiplier before making your final determination about the inequality symbol. This step is the culmination of all our efforts, revealing the ultimate condition that 'x' must satisfy. The careful application of inverse operations, combined with the golden rule of inequalities regarding sign flipping, brings us to our precise and accurate solution for 'x'. It's the moment of truth, where all the previous simplifications lead to a clear and unambiguous statement about the variable.
Understanding the Solution Set for x < 12
Alright, so we've done the math, and we've landed on x < 12. But what does that actually mean? This isn't just one number; it's a whole bunch of numbers, guys! The solution set for x < 12 includes any and every real number that is strictly less than 12. This means numbers like 11, 10, 0, -5, -100, 11.9, 11.999 – you get the idea! As long as the number is smaller than 12, it's part of our solution. What's not included? Well, 12 itself isn't part of the solution, because 'x' has to be less than 12, not less than or equal to. Numbers like 12.01, 13, 100? Nope, they're too big. Think of it on a number line. If you were to draw this, you'd put an open circle (or a hollow circle) at 12 to show that 12 is not included. Then, you'd draw an arrow extending to the left from 12, indicating that all numbers in that direction (towards negative infinity) are part of the solution. In fancy math terms, we often write this solution set using interval notation as (-∞, 12). The parenthesis next to 12 means 12 is excluded, and negative infinity (always with a parenthesis) shows that the numbers go on forever in the negative direction. It's a comprehensive way to express all the possible values of 'x' that satisfy the initial condition. Understanding this graphical and interval representation is just as important as solving the inequality itself, as it provides a complete picture of the solution. It’s not just about a single number, but an entire continuum of values, which makes inequalities so powerful in describing ranges and boundaries in various fields from science to economics. This interpretation is the final piece of the puzzle, transforming a mathematical statement into a clear and visually understandable concept. It's about grasping the scope of the solution rather than just its final numerical form, ensuring you fully comprehend what x < 12 truly represents in the grand scheme of numbers.
Why Other Options Aren't Right (and Common Pitfalls)
Let's quickly address why the other choices you might see (like x > 12, x < 16, or x > 16) are incorrect for our specific inequality, 5(x+5) < 85. This also gives us a chance to highlight some common pitfalls that many people stumble upon. If you had ended up with x > 12, it means you likely made an error in isolating 'x' or perhaps incorrectly flipped the inequality sign somewhere along the way. For instance, if you had accidentally divided by a negative number in your calculations (which we didn't, but it's a common hypothetical scenario), that's where the sign might have flipped incorrectly. Similarly, if your calculations led to x < 16 or x > 16, it points to an arithmetic mistake, perhaps in the distribution step (5 multiplied by 5 should be 25, not something else) or in the subtraction step (85 - 25 should be 60, not a different number). These seemingly small calculation errors can lead to entirely different solution sets. Another common mistake is forgetting to distribute the '5' to both the 'x' and the '5' inside the parentheses; if you only multiplied 5 by x and left the +5 untouched, your equation would have looked very different. Always, always double-check your basic arithmetic and ensure you're applying the distributive property correctly. These little details are where inequalities can get tricky, but by being meticulous and reviewing each step, you can easily avoid these common blunders. Remember, the journey to the correct answer is just as important as the answer itself, and understanding where mistakes can happen is a powerful learning tool. So, if your answer didn't match x < 12, take a deep breath, retrace your steps, and see where the path diverged. It's all part of the learning process, and identifying those missteps is a crucial part of mastering inequalities. By understanding the common errors, you're better equipped to catch them in your own work and in the work of others, solidifying your understanding of the process and making you a more skilled problem-solver. It’s like having a cheat sheet for avoiding traps in a video game – you know where to look and what to avoid to successfully complete your mission.