Solving Compound Inequalities: A Step-by-Step Guide

Hey guys! Let's dive into the world of compound inequalities and break down how to solve them. This guide will help you understand how to tackle problems like the one you've got, which is "Which compound inequality represents the inequality $3|y+7|-16>5$ ?" Don't worry; it's not as scary as it looks. We'll go through the steps, and by the end, you'll be a compound inequality whiz! This is a fundamental concept in algebra, and understanding it is crucial for more advanced math topics. So, grab your pencils, and let's get started!

Understanding the Basics of Compound Inequalities

Alright, before we jump into the specific problem, let's get a handle on what compound inequalities are all about. Basically, a compound inequality is an inequality that combines two simple inequalities. There are two main types: "and" inequalities and "or" inequalities. "And" inequalities mean that both conditions must be true for the inequality to be satisfied. Think of it like this: You need to be both tall and have a good sense of humor to be a basketball player. Both conditions are required. "Or" inequalities, on the other hand, mean that at least one of the conditions must be true. If you're either a cat or a dog, you're a pet. Only one condition is required. Now, the question you've got involves absolute values, which adds another layer to our understanding. The absolute value of a number is its distance from zero, so it's always non-negative. When we deal with absolute value inequalities, we'll usually end up with two separate inequalities to solve, reflecting the two possible scenarios due to the absolute value. Let's break down our example and see how this all comes together, step by step, so you can master these problems. Keep in mind, compound inequalities are building blocks for more complex math concepts, so understanding this is super important!

Step-by-Step Solution for $3|y+7|-16>5$

Okay, let's get down to business and solve the inequality $3|y+7|-16>5$. Here's how we'll do it, step-by-step: Step 1: Isolate the Absolute Value. First, we want to get the absolute value expression by itself on one side of the inequality. We'll do this by adding 16 to both sides of the inequality to get rid of that -16. This gives us: $3|y+7| > 21$. Next, we divide both sides by 3 to isolate the absolute value completely: $|y+7| > 7$. Step 2: Break it Down. Now, here's where the magic happens. Because we have an absolute value, we need to consider two separate cases. Remember, the absolute value represents the distance from zero, so both positive and negative values inside the absolute value brackets are possible. This means we need to consider both $y+7 > 7$ and $-(y+7) > 7$ (which simplifies to $y+7 < -7$). Essentially, we're saying that the expression inside the absolute value can be either greater than 7 or less than -7. This is where the "or" comes in. Step 3: Solve the Inequalities. Now we solve each of the two inequalities separately. For $y+7 > 7$, subtract 7 from both sides, and we get $y > 0$. For $y+7 < -7$, subtract 7 from both sides, and we get $y < -14$. Step 4: Combine and Interpret. Finally, we combine these two inequalities. Since we used "or" in our thought process when splitting the absolute value, we use "or" to combine our solutions. Therefore, the compound inequality that represents the original inequality is $y < -14$ OR $y > 0$. This means any value of y that is less than -14 or greater than 0 satisfies the original inequality. That means the correct answer among the options provided is B: $y+7<-7$ OR $y+7>7$ because they are mathematically equivalent to $y<-14$ OR $y>0$.

Understanding the Options

Let's take a quick look at the multiple-choice options to make sure we understand why the correct answer is correct, and the others are not. We've already determined the correct answer through our step-by-step solution, but a quick review is helpful. Option A: $y+7>-7$ AND $y+7<7$. This corresponds to $-7 < y+7 < 7$, which means $-14 < y < 0$. This is incorrect because it represents an "and" condition and a different range of values compared to what we found. Option C: $y+7<-7$ AND $y+7>7$. This simplifies to $y < -14$ AND $y > 0$. This is incorrect because there are no values of y that can be both less than -14 and greater than 0 simultaneously. This is a contradiction. Option D: $y+7>-7$ OR $y+7<7$. This simplifies to $y > -14$ OR $y < 0$. This is close, but the inequality signs are not quite right. This option combines the results of $y+7>7$ OR $y+7<-7$, which translates to $y>-14$ OR $y<0$. This is the closest, but still incorrect since it's not the same as the correct answer. By carefully analyzing each option, we can see why our step-by-step solution leads us to the right answer. This process of breaking down the problem, solving it, and then matching it to the correct answer is a solid strategy for any multiple-choice question.

Tips for Solving Compound Inequalities

Here are a few tips to help you ace compound inequality problems: Tip 1: Isolate First. Always isolate the absolute value expression (if there is one) before dealing with the "and" or "or" conditions. This simplifies the problem. Tip 2: Remember the Rules. If the absolute value is greater than a number, you use "or". If the absolute value is less than a number, you use "and". This is super important! Tip 3: Double-Check Your Signs. When splitting the absolute value, be extra careful with the signs. It's easy to make a mistake here. Tip 4: Visualize the Solution. Draw a number line to visualize the solution, especially if you're dealing with "and" or "or" inequalities. This helps you see the range of values that satisfy the inequality. Tip 5: Practice, Practice, Practice. The more problems you solve, the better you'll get at recognizing patterns and solving them quickly. Doing lots of problems will build your confidence and make you feel comfortable with the concepts. Keep these tips in mind, and you'll be well on your way to becoming a compound inequality pro! Remember, understanding the steps and practicing consistently is the key to success in math.

Conclusion

So, there you have it! We've successfully solved the compound inequality $3|y+7|-16>5$. We walked through the steps of isolating the absolute value, breaking it down into two separate inequalities, solving them, and combining the results. By understanding the basics of compound inequalities, remembering the rules for absolute values, and following a systematic approach, you can tackle these problems with confidence. Keep practicing, and you'll find yourself becoming more and more comfortable with these types of problems. You got this! Keep up the great work, and always remember to go back and check your work. Good luck with your future math adventures!