Solving Inequalities: A Step-by-Step Guide

Hey guys! Let's dive into solving inequalities. It might seem tricky at first, but trust me, it's super manageable once you break it down. We'll take the inequality 9h + 2 < -79 as our example and walk through each step together. So, buckle up, and let's get started!

Understanding Inequalities

Before we jump into the problem, let's quickly recap what inequalities are all about. Unlike equations that have a single solution, inequalities deal with a range of solutions. Think of it as finding all the values that make a statement true within a certain range. Inequalities use symbols like < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). When we're solving inequalities, our main goal is to isolate the variable, just like we do with equations. However, there's one crucial difference: if we multiply or divide by a negative number, we need to flip the inequality sign. Keep this in mind, as it's a common pitfall for many. Mastering this concept is key to accurately solving inequalities. The ultimate goal is to simplify the inequality to a point where the variable is isolated on one side, revealing the range of values that satisfy the condition. This is usually represented as something like x < a or x > b, where a and b are constants.

Why Inequalities Matter

You might be wondering, why bother with inequalities anyway? Well, they pop up everywhere in real life! From figuring out budget constraints to optimizing resources, inequalities help us model and solve problems where there's a range of possible solutions. For example, imagine you're planning a party and have a budget. You need to make sure your expenses (food, drinks, decorations) stay below a certain amount. This is a perfect scenario where inequalities come into play. Or, think about speed limits on a road. The car's speed must be less than or equal to the posted limit. In business, companies use inequalities to determine how much they need to sell to make a profit, ensuring their costs are less than their revenue. In science, inequalities can help define the range of acceptable values for an experiment to ensure accuracy. So, you see, understanding inequalities is not just about math class; it's a valuable skill for navigating many aspects of life. The ability to translate real-world scenarios into mathematical inequalities and then solve them gives you a powerful tool for decision-making and problem-solving.

Common Pitfalls to Avoid

Alright, let's talk about some common mistakes people make when solving inequalities. The biggest one, as I mentioned earlier, is forgetting to flip the inequality sign when multiplying or dividing by a negative number. It's super important to remember this rule! Another mistake is not distributing correctly when dealing with expressions in parentheses. Make sure you multiply the number outside the parentheses by every term inside. Also, pay close attention to the direction of the inequality sign. A simple mix-up can lead to a completely wrong answer. For instance, if you end up with h > -10 but the correct answer is h < -10, you've missed a critical detail. Lastly, be careful when dealing with compound inequalities (like a < x < b). These inequalities represent a range between two values, so you need to handle each part of the inequality correctly. Practice makes perfect, so the more you work with inequalities, the easier it'll become to avoid these pitfalls. And remember, double-checking your work is always a good idea, especially in math! Catching errors early on can save you from a lot of frustration down the line.

Step-by-Step Solution: 9h + 2 < -79

Okay, now let's tackle our example inequality: 9h + 2 < -79. We're going to break it down into easy-to-follow steps. Don't worry, it's not as scary as it looks!

Step 1: Isolate the Term with the Variable

Our first mission is to get the term with the variable (in this case, 9h) by itself on one side of the inequality. To do this, we need to get rid of that +2. How do we do it? We use the inverse operation, which is subtraction. We'll subtract 2 from both sides of the inequality. This keeps the inequality balanced, just like with equations. So, we have:

9h + 2 - 2 < -79 - 2

This simplifies to:

9h < -81

See? We're already one step closer! This step is all about undoing the addition or subtraction that's cluttering the variable term. It's like peeling back the layers of an onion – we're getting to the core of the problem. Remember, whatever you do to one side of the inequality, you must do to the other side to maintain the balance. This principle is fundamental to solving any algebraic equation or inequality.

Step 2: Isolate the Variable

Now we have 9h < -81. The variable 'h' is almost isolated, but it's still being multiplied by 9. To get 'h' completely alone, we need to undo this multiplication. The inverse operation of multiplication is division, so we'll divide both sides of the inequality by 9. Here's what it looks like:

(9h) / 9 < (-81) / 9

This simplifies to:

h < -9

And there we have it! We've solved the inequality. This step is crucial because it directly reveals the solution set. Dividing (or multiplying) is the final action required to free the variable and see the range of values that satisfy the inequality. You've successfully navigated to the final step by correctly applying the inverse operation. In this specific case, dividing both sides by 9 was the key to unveiling the solution.

Step 3: Interpret the Solution

Our solution is h < -9. What does this mean? It means that any value of 'h' that is less than -9 will make the original inequality true. Think of it as a range of solutions, not just one single answer. For example, -10, -11, -100, even -9.0001 all satisfy this inequality. To truly grasp the solution, it's helpful to visualize it on a number line. Imagine a number line stretching from negative infinity to positive infinity. Locate -9 on the line. Because our solution is 'h less than -9' (and not 'less than or equal to'), we'll draw an open circle at -9, indicating that -9 itself is not included in the solution. Then, we'll shade everything to the left of -9, representing all the numbers that are less than -9. This visual representation provides a clear and intuitive understanding of the solution set. It shows the infinite range of values that make the inequality true, emphasizing the difference between 'less than' and 'less than or equal to'.

Checking Your Answer

It's always a good idea to check your answer, guys! This way, you can be super confident that you've got it right. To check our solution (h < -9), we can pick a number that's less than -9 and plug it back into the original inequality. Let's choose -10 (since it's a nice, easy number). So, we'll substitute h = -10 into 9h + 2 < -79:

9(-10) + 2 < -79

-90 + 2 < -79

-88 < -79

Is -88 less than -79? Yes, it is! This confirms that our solution h < -9 is correct. If we had gotten a false statement, like -88 > -79, it would mean we made a mistake somewhere, and we'd need to go back and check our steps. Checking your answer is a critical step in the problem-solving process. It gives you a safety net, ensuring that your solution is not only mathematically correct but also logically sound. By plugging in a test value, you're essentially verifying that your answer fits the conditions set by the original inequality.

Practice Makes Perfect

Solving inequalities is a skill that gets better with practice. The more you do it, the more comfortable you'll become with the steps and the tricky parts. Try solving different inequalities with varying levels of complexity. Start with simple ones like x + 3 < 7, and then move on to more challenging ones involving parentheses, fractions, and negative numbers. Remember to always show your work and double-check your answers. You can find tons of practice problems online or in textbooks. Look for resources that provide step-by-step solutions so you can see where you might be going wrong. And don't be afraid to ask for help! If you're stuck on a problem, reach out to a teacher, tutor, or classmate. There are also many online forums and communities where you can ask questions and get support. The key is to actively engage with the material and not just passively read through it. Solving problems on your own, even if it takes time and effort, is the best way to truly understand the concepts. So, keep practicing, and you'll become an inequality-solving pro in no time!

So, the correct answer for 9h + 2 < -79 is B. h < -10. You did it!

Remember, math is like any other skill – the more you practice, the better you get. Keep up the awesome work, and you'll be conquering those inequalities in no time!