Solving Compound Inequalities Who Is Correct Sophie, Adola, Or Gilly?

Hey guys! Let's dive into a fun math problem today where we're going to figure out who's right among Sophie, Adola, and Gilly. They're debating about the solutions to a compound inequality. Compound inequalities might sound intimidating, but trust me, they're totally manageable once you break them down. We'll explore the problem step by step, making sure everyone understands the process. So, grab your thinking caps, and let’s get started!

Understanding the Problem

Okay, so here’s the deal. Sophie, Adola, and Gilly are trying to find values that fit into a particular compound inequality: $-2 \leq 2x - 2 < 8$. A compound inequality is basically two inequalities joined together. In this case, we have $-2$ is less than or equal to $2x - 2$, and $2x - 2$ is less than $8$. The key here is that both parts of the inequality must be true for a number to be a solution. Sophie thinks -2 works, Adola is betting on 5, and Gilly is championing 2. Our mission? To figure out which one of them, if any, is correct. We need to test each number in the inequality and see if it holds true. So, let's get to it!

Breaking Down the Compound Inequality

Before we jump into testing the numbers, let’s take a moment to really understand what this compound inequality is telling us. Think of it as a set of rules that the variable x has to follow. The first rule is that when you plug x into the expression $2x - 2$, the result must be greater than or equal to $-2$. The second rule is that the same expression, $2x - 2$, must also be less than $8$. It’s like a balancing act – the value of $2x - 2$ has to fall within a certain range. To make things clearer, we can actually solve this compound inequality to find all possible values of x. This involves isolating x in the middle, which means we need to perform the same operations on all three parts of the inequality. Remember, whatever we do to one part, we have to do to all parts to keep the inequality balanced. This will give us a range of values that x can take, and then we can easily see if Sophie, Adola, and Gilly's suggestions fit within that range.

Solving for x

Alright, let's get our hands dirty and solve this compound inequality! Our goal is to isolate x in the middle, so we need to get rid of the $-2$ and the $2$ that are hanging around with it. The first step is to get rid of the $-2$ that's being subtracted. To do that, we'll add $2$ to all three parts of the inequality. This gives us: $-2 + 2 \leq 2x - 2 + 2 < 8 + 2$, which simplifies to $0 \leq 2x < 10$. Great! We're one step closer. Now, we need to get rid of the $2$ that's multiplying x. To do that, we'll divide all three parts of the inequality by $2$. This gives us: $0 / 2 \leq 2x / 2 < 10 / 2$, which simplifies to $0 \leq x < 5$. So, we've found our solution! This inequality tells us that x can be any number greater than or equal to $0$, but it must be strictly less than $5$. In other words, x can be $0$, $1$, $2$, $3$, $4$, and all the decimals in between, but it cannot be $5$ or any number greater than $5$. Now that we know the range of possible values for x, we can check Sophie, Adola, and Gilly's suggestions.

Testing Sophie's Suggestion: -2

Let’s start with Sophie, who believes that -2 is a solution. Remember, our compound inequality is $-2 \leq 2x - 2 < 8$. To check if -2 is a solution, we need to plug it in for x and see if both parts of the inequality hold true. So, we substitute -2 for x: $-2 \leq 2(-2) - 2 < 8$. Now, let's simplify the middle part: $2(-2) - 2 = -4 - 2 = -6$. So, our inequality becomes $-2 \leq -6 < 8$. Now, let’s analyze this. Is $-2$ less than or equal to $-6$? No, it’s not! $-2$ is actually greater than $-6$. So, the first part of the compound inequality is false. We don't even need to check the second part because for a number to be a solution to a compound inequality, both parts must be true. Since the first part is false, we can confidently say that -2 is not a solution. Sorry, Sophie!

The Verdict on Sophie's Suggestion

So, we plugged -2 into our compound inequality, and it didn't hold up. The first part of the inequality, $-2 \leq 2x - 2$, became $-2 \leq -6$, which is definitely not true. Since one part of the compound inequality is false, the entire statement is false. This means that -2 is not a solution to the inequality. You see, with compound inequalities, it's like having two locks on a door. You need both keys to open it. In this case, the two keys are the two inequalities that make up the compound inequality. If one of the inequalities isn't satisfied, then the number doesn't work as a solution. So, Sophie's suggestion doesn't fit the bill. But don't worry, we still have Adola and Gilly's suggestions to check. Let's see who comes closer to the right answer!

Examining Adola's Claim: 5

Next up is Adola, who thinks 5 is a solution to our compound inequality, $-2 \leq 2x - 2 < 8$. Just like we did with Sophie's suggestion, we need to substitute 5 for x and see if the inequality holds true. Plugging in 5, we get: $-2 \leq 2(5) - 2 < 8$. Now, let’s simplify the middle part: $2(5) - 2 = 10 - 2 = 8$. So, our inequality becomes $-2 \leq 8 < 8$. Let's break this down. Is $-2$ less than or equal to $8$? Yes, it is! So, the first part of the inequality is true. But what about the second part? Is $8$ less than $8$? Nope, it's not! $8$ is equal to $8$, but it's not less than $8$. This is a crucial point to remember about inequalities: the symbol “<” means strictly less than. Since the second part of the compound inequality is false, the entire statement is false. Therefore, 5 is not a solution. Nice try, Adola, but let’s see if Gilly fares any better!

Why 5 Doesn't Work

Let’s really understand why 5 doesn't work as a solution. When we plugged 5 into the compound inequality, we ended up with $-2 \leq 8 < 8$. The first part, $-2 \leq 8$, is true – no problem there. But the second part, $8 < 8$, is the troublemaker. Remember, the “less than” symbol (<) means a number has to be strictly smaller. It can't be equal to. So, 8 cannot be less than itself. This is a common mistake people make when working with inequalities, so it’s a good one to understand thoroughly. Think of it like this: if you have 8 cookies, you don’t have less than 8 cookies; you have exactly 8 cookies. Since this second part of the compound inequality is false, the whole thing falls apart. 5 is right on the edge of being a solution, but it just misses the mark because of that strict “less than” requirement. Okay, one down, one more to go. Let's see if Gilly can pull through!

Evaluating Gilly's Choice: 2

Alright, last but not least, we have Gilly, who suggests that 2 is a solution to the compound inequality, $-2 \leq 2x - 2 < 8$. By now, we know the drill! We’re going to substitute 2 for x and check if the inequality holds true. So, let's plug it in: $-2 \leq 2(2) - 2 < 8$. Now, let's simplify the middle part: $2(2) - 2 = 4 - 2 = 2$. This gives us the inequality $-2 \leq 2 < 8$. Time to analyze! Is $-2$ less than or equal to $2$? Yes, it is! So, the first part of the inequality is true. And what about the second part? Is $2$ less than $8$? Absolutely! So, the second part is also true. Since both parts of the compound inequality are true, we have a winner! Gilly is correct; 2 is indeed a solution to the compound inequality. Woohoo, Gilly!

Gilly's Triumph: Why 2 Works

Let's celebrate Gilly's victory and understand why 2 is the correct answer. When we substituted 2 into the compound inequality, $-2 \leq 2x - 2 < 8$, we ended up with $-2 \leq 2 < 8$. Both parts of this statement are true: $-2$ is less than or equal to $2$, and $2$ is less than $8$. This is exactly what we need for a number to be a solution to a compound inequality. It has to satisfy both conditions. Think back to our analogy of the two locks on a door. Gilly's suggestion, 2, had both the keys! It unlocked both inequalities, making it a valid solution. Remember when we solved the inequality and found that $0 \leq x < 5$? Well, 2 falls perfectly within that range. It's greater than or equal to 0, and it's less than 5. So, Gilly not only got the right answer, but it also makes sense in the context of the solution set we found earlier. Great job, Gilly!

Final Verdict: Who is Correct?

After carefully testing each suggestion, we’ve reached our final verdict! Sophie thought -2 was a solution, but it didn't satisfy the compound inequality. Adola believed 5 was the answer, but it fell short because it didn't meet the “less than” requirement. And finally, Gilly correctly identified 2 as a solution. So, the winner is… Gilly! Gilly understood how to check if a number is a solution to a compound inequality by substituting it into the inequality and verifying that both parts of the statement hold true. This problem highlights the importance of paying close attention to the details of inequalities, like the difference between “less than or equal to” and “less than.” Understanding these nuances is key to solving these types of problems accurately. So, let's give Gilly a virtual round of applause for acing this challenge!

Key Takeaways and Tips

Before we wrap up, let’s quickly recap some key takeaways from this problem. First, remember what a compound inequality is: it’s two inequalities joined together, and both parts must be true for a number to be a solution. Second, when checking if a number is a solution, substitute it into the inequality and simplify. Then, carefully analyze each part of the resulting statement. Third, pay close attention to the inequality symbols. “Less than” (<) means strictly less than, while “less than or equal to” ($\leq$) includes the possibility of equality. Finally, if you're unsure, solve the inequality first to find the range of possible solutions. This can make it much easier to check individual values. Solving inequalities is a fundamental skill in math, and with practice, you’ll become a pro at it! Keep practicing, and you'll be solving compound inequalities like a champ in no time.

Wrapping Up

Well, guys, we’ve had quite the mathematical adventure today! We successfully navigated a compound inequality, tested different suggestions, and crowned Gilly as our champion solver. Hopefully, this exercise has not only helped you understand how to solve compound inequalities but also shown you how to approach problems systematically and think critically. Remember, math is all about understanding the rules and applying them correctly. So, keep exploring, keep questioning, and most importantly, keep practicing! You've got this!