Solving 9|x-8|<36 A Step-by-Step Guide

Hey guys! 👋 Let's dive into solving this inequality question together. Inequalities can sometimes seem tricky, but don't worry, we'll break it down step by step. Our main goal here is to figure out the solution to the inequality $9|x-8|<36$. This involves understanding absolute values and how they affect our solutions. So, grab your thinking caps, and let's get started!

Understanding Absolute Value Inequalities

Alright, first things first, let’s tackle absolute value inequalities. What exactly does that mean? Well, the absolute value of a number is its distance from zero on the number line. Think of it this way: $|5|$ is 5 because 5 is five units away from zero. Similarly, $|-5|$ is also 5 because -5 is also five units away from zero. This concept is super important when we're dealing with inequalities.

When you see an absolute value inequality like $|x| < a$, it means we're looking for all the values of $x$ that are less than $a$ units away from zero. This translates into two separate inequalities: $-a < x < a$. On the flip side, if we have $|x| > a$, it means we want all the values of $x$ that are more than $a$ units away from zero. This splits into two inequalities as well: $x < -a$ or $x > a$. Knowing these rules is like having a secret key to unlock these types of problems.

Now, let’s bring this back to our problem: $9|x-8|<36$. Notice the absolute value part $|x-8|$. This is where things get interesting. The expression inside the absolute value, $x-8$, is what we need to focus on. It represents the distance between $x$ and 8 on the number line. We want this distance, when multiplied by 9, to be less than 36. So, to really nail this, we've got to isolate the absolute value first. This involves a bit of algebraic maneuvering, but trust me, it's totally manageable.

The idea here is to get the absolute value expression by itself on one side of the inequality. Once we’ve done that, we can apply the rules we just talked about to split the inequality into two separate, more manageable inequalities. This is where the problem starts to look less intimidating and more like a puzzle we can solve. So, stick with me, and let’s break down the steps one by one. We're not just solving a problem here; we're building a solid understanding of how absolute value inequalities work. This is a skill you'll use again and again in math, so it’s worth getting it right!

Step-by-Step Solution for 9|x-8|<36

Okay, let's break down the solution to $9|x-8|<36$ step-by-step. First up, we need to isolate the absolute value. Remember, we want to get $|x-8|$ all by itself on one side of the inequality. To do that, we've got to get rid of that 9 that’s hanging out in front.

So, how do we do it? Simple! We divide both sides of the inequality by 9. This is a classic algebraic move. What we do to one side, we have to do to the other to keep things balanced. When we divide both sides by 9, we get:

$|x-8| < 4$

See? That's much cleaner already! Now we've got the absolute value isolated, which means we're ready for the next big step: splitting this into two separate inequalities. Remember the rule we talked about earlier? When you have an absolute value inequality in the form $|x| < a$, it turns into $-a < x < a$. We're going to apply that same logic here.

Our absolute value expression is $|x-8|$, and it's less than 4. So, we can rewrite this as:

$-4 < x-8 < 4$

Now we've got a compound inequality. It looks a bit more complex, but don't sweat it. It's just two inequalities smooshed together, and we can solve it pretty easily. Our goal now is to isolate $x$ in the middle. To do that, we need to get rid of the -8. The way we do that is by adding 8 to all parts of the inequality. Yep, that means the left side, the middle, and the right side. When we add 8 to all parts, we get:

$-4 + 8 < x - 8 + 8 < 4 + 8$

Simplifying that, we have:

$4 < x < 12$

And there you have it! We've solved the inequality. This means that $x$ is greater than 4 but less than 12. We can also write this in interval notation as $(4, 12)$. This is the range of values that $x$ can take to make the original inequality true. See how breaking it down step-by-step makes it much more manageable? We isolated the absolute value, split the inequality, and then solved for $x$. Each step is logical and builds on the previous one. This is the key to mastering inequalities!

Analyzing the Answer Choices

Alright, now that we've got our solution, $4 < x < 12$, let's take a look at the answer choices and see which one matches up. This is a crucial step because sometimes the answer might be presented in a slightly different way than we expect.

We've got our solution: $4 < x < 12$. This means $x$ is stuck between 4 and 12. It's bigger than 4, but it's also smaller than 12. Think of it like a sweet spot on a number line. We're looking for an answer choice that says exactly that. We can visualize this on a number line. Imagine a line with numbers stretching out in both directions. We'd put an open circle at 4 because $x$ is greater than 4 but doesn't include 4 itself. Then we'd put another open circle at 12 for the same reason. Finally, we'd shade the line in between those two circles to show all the possible values of $x$.

Now, let's go through those answer choices one by one. We're looking for the one that correctly describes this sweet spot between 4 and 12:

A. $4$: This is just a single value, and we need a range of values. So, A is not our answer.

B. $x<-4$ or $x>12$: This says $x$ is either less than -4 or greater than 12. That's the opposite of what we want! We want $x$ to be between 4 and 12, not outside those numbers. So, B is definitely not it.

C. $x>-12$ or $x<8$: This one is tricky because it uses "or," but it doesn't quite match our solution. It says $x$ is greater than -12, which is true in our case, but it also says $x$ is less than 8, which isn't entirely accurate. Our solution is more specific: $x$ is less than 12. So, C isn't the best fit.

D. $-4 < x < 12$: This looks promising! It says $x$ is greater than -4 and less than 12. Wait a second… we found $4 < x < 12$. But let's think about this. Is every number between 4 and 12 also greater than -4? Yep! So, this answer choice captures the upper bound correctly ($x < 12$), but the lower bound is a bit broader than our exact solution. This could be a tricky one designed to make you think twice.

However, after careful consideration, we realize that there seems to be a slight error in the provided options. None of them perfectly match our solution of $4 < x < 12$. The closest one is D, but it includes values less than 4, which are not part of our solution. It's essential to recognize when answer choices might be flawed and to stick to the correct solution we derived.

Common Mistakes and How to Avoid Them

Alright, let's talk about some common pitfalls people often stumble into when solving inequalities, especially those involving absolute values. Knowing these mistakes can help you dodge them and nail these problems every time!

One biggie is forgetting to split the absolute value inequality into two separate inequalities. Remember, an absolute value expression can represent distances in both positive and negative directions from zero. So, when you see $|x| < a$, it’s not just about $x < a$; you also have to consider $x > -a$. Missing this step is like only seeing half the picture. Always remember to split the inequality to cover both possibilities. In our problem, failing to split $9|x-8|<36$ into two cases would lead to an incomplete or incorrect solution.

Another common mistake is messing up the inequality signs when dealing with negative numbers. This is a classic algebra gotcha. When you multiply or divide both sides of an inequality by a negative number, you've got to flip the inequality sign. For example, if you have $-x < 5$, you need to divide by -1 to solve for $x$, but you also need to flip the sign, giving you $x > -5$. Forgetting this flip can completely change your answer. While this specific issue didn't come up directly in our problem, it's a crucial rule to remember for other inequality scenarios.

Also, watch out for arithmetic errors! It sounds basic, but simple calculation mistakes can throw off your entire solution. Always double-check your work, especially when adding, subtracting, multiplying, or dividing. It's super easy to make a small slip, but those slips can lead to wrong answers. In our problem, if we had made an arithmetic error while dividing by 9 or adding 8, we would have ended up with the wrong range for $x$.

Lastly, be careful with the direction of the inequality signs. It's easy to get mixed up between "less than" and "greater than." Make sure you're reading the signs correctly and representing the solution accurately. A great way to double-check is to plug a value from your solution back into the original inequality to see if it holds true. If it doesn't, something went wrong somewhere, and it's time to retrace your steps.

Real-World Applications of Inequalities

Okay, so we've conquered the math, but let's take a step back and think about why this stuff matters in the real world. Inequalities aren't just abstract math problems; they pop up in all sorts of everyday situations. Understanding them can actually be super useful!

Think about budgeting, for example. Let’s say you've got a certain amount of money to spend each month. You can use an inequality to represent your spending limits. If you want to make sure your expenses are less than or equal to your income, you're essentially setting up an inequality. This helps you keep your finances in check and avoid overspending.

Inequalities are also crucial in science and engineering. Imagine you're designing a bridge. You need to ensure the materials you're using can withstand a certain amount of weight. You'd use inequalities to define the maximum stress the bridge can handle. Or, in chemistry, you might use inequalities to describe the range of temperatures at which a reaction will occur safely. In fact, quality control often uses inequalities to define acceptable ranges for product dimensions or performance metrics. If a product falls outside these ranges, it doesn't meet the required standards.

They're used in setting speed limits on roads, defining the minimum and maximum doses of medications, and even determining eligibility criteria for scholarships or loans. Ever wondered how websites handle password requirements? They often use inequalities to ensure passwords meet minimum length and complexity criteria.

In computer science, inequalities are used in algorithms to optimize performance. For example, they might be used to determine the minimum number of resources needed to run a program efficiently or the maximum amount of time an algorithm can take to complete. In the world of business and economics, inequalities are used in optimization problems, such as maximizing profits or minimizing costs. For example, a company might use linear programming (which relies heavily on inequalities) to determine the optimal production levels for its products.

The point is, inequalities are way more than just symbols on a page. They're a fundamental tool for making decisions, setting limits, and solving problems across a huge range of fields. So, by mastering inequalities, you're not just acing your math test; you're also building a skill that will serve you well in the real world. Keep practicing, keep exploring, and you'll be amazed at how useful inequalities can be!

So, to wrap it all up, we tackled the inequality $9|x-8|<36$. We broke it down step by step, from isolating the absolute value to splitting the inequality and solving for $x$. We discovered that the solution is $4 < x < 12$. While none of the provided answer choices perfectly matched our solution, we learned the importance of sticking to our derived answer and recognizing potential errors in the options. Remember, math isn't just about getting the right answer; it's about understanding the process. Keep practicing, and you'll become an inequality-solving pro in no time!