Deciphering Inequalities: Finding Positive Values When A < 0

Hey guys! Let's dive into a classic math problem that tests our understanding of inequalities. This type of question is super common in algebra, and mastering it will give you a solid foundation for more complex concepts. So, the question starts by giving us a crucial piece of information: a < 0. This means 'a' is a negative number. Our mission is to figure out which of the given expressions will always result in a value greater than zero (positive), no matter what specific negative value 'a' might be. Sounds fun, right?

This kind of problem is all about testing your ability to work with inequalities. It's not about doing heavy calculations; it's more about logical reasoning and understanding how different operations affect the sign (positive or negative) of a number. Here's how we're going to break it down. We'll examine each expression, thinking through the possible scenarios, and keeping in mind that 'a' is negative. We're looking for the expression that guarantees a positive result. This requires careful consideration of the signs of the numbers involved and how they interact during addition, subtraction, division, and multiplication. Are you ready to crack this code? Let's get started!

Understanding the Basics: Negative Numbers and Inequalities

Alright, before we get our hands dirty with the expressions, let's refresh our memory on the basics of working with negative numbers and inequalities. When we say a < 0, we're saying that 'a' is less than zero, meaning it's a negative number. For example, 'a' could be -1, -5, or -100—any number that's on the left side of zero on the number line. Now, what happens when we perform operations like addition, subtraction, and division with these negative numbers? That's what we need to consider.

First, think about addition and subtraction. When you add or subtract a negative number, you're essentially moving along the number line. Adding a negative number is like subtracting, and subtracting a negative number is like adding. The key is to keep track of the signs. For instance, if you have a negative number and subtract another negative number (which is like adding a positive), you might end up with a positive result, depending on the numbers involved. On the other hand, adding a negative number to a negative number will result in a more negative number. When dealing with multiplication and division, there's a simple rule: if the signs are the same (both positive or both negative), the result is positive. If the signs are different (one positive and one negative), the result is negative.

So, understanding the behavior of signs during these operations is key to solving this problem. Keep in mind that when we're trying to find an expression that's always positive, we need to make sure that the operations consistently lead to a positive result, regardless of the specific values of the other variables involved. We must, therefore, methodically analyze each option to determine if it guarantees a positive outcome in light of the fact that 'a' is negative. Let's see what we can do, shall we?

Analyzing the Expressions: Finding the Always Positive Value

Now, let's get into the heart of the problem. We'll take each of the provided expressions one by one and try to figure out whether it always yields a positive result when a < 0. Remember, we are not trying to calculate the exact values, but rather assess if the outcome is always positive, regardless of the specific numbers used. This is where your logical reasoning skills come in handy!

Expression A: (a - c) / (c + b)

With this expression, we have a subtraction in the numerator (a - c) and an addition in the denominator (c + b). Since we only know that 'a' is negative, we have no information about 'b' and 'c'. That means the sign of the numerator and denominator could change depending on the values of 'b' and 'c'. For example, if 'c' is a large positive number, then (a - c) will be negative (a negative minus a positive is always negative), and (c + b) could be positive or negative depending on 'b'. The ratio might be positive, but it is not guaranteed. Hence, this option is not our pick.

Expression B: (a - b) / (c - a)

Here, we also have subtraction in the numerator and denominator. We know 'a' is negative, but we don't have information about 'b' and 'c'. If 'b' is a positive number and 'a' is negative, (a - b) will certainly be negative. In the denominator, we have (c - a). Since 'a' is negative, then subtracting a negative number is like adding a positive number. If 'c' is also negative, then (c - a) could potentially be positive if the magnitude of 'a' is greater than 'c'. This makes this choice a possible pick; however, it is not always guaranteed. We keep searching!

Expression C: (a - c) / (b - c)

Again, we have subtraction in both the numerator and denominator. The numerator is (a - c). Since 'a' is negative and we have no information on 'c', the result can vary. The denominator is (b - c), which can also take a range of values based on what 'b' and 'c' are. This expression does not provide a guaranteed positive result.

Expression D: (a + b) / (c + a)

In this expression, we have (a + b) and (c + a). We know 'a' is negative. Since 'a' is negative, the denominator (c + a) is less than 'c', and the numerator (a + b) is less than 'b'. If we consider the case where 'b' is a positive number and has a magnitude greater than that of 'a', then (a + b) will be positive, and (c + a) can be positive or negative depending on 'c'. However, the value is not always guaranteed to be positive. If we consider 'b' and 'c' as negative, then it is possible to achieve an always-positive outcome. Let's consider some examples: Let's assume a = -2, b = -3, and c = -4. In this case, (a + b) is -5, and (c + a) is -6, and thus the overall result would be positive! Does it mean that this expression is the correct answer? Not yet! We have to think of other scenarios.

To make this expression always positive, the only way is to have (a + b) as negative and (c + a) as negative, or (a + b) as positive and (c + a) as positive. If we assume that b and c are negative, then (a + b) will always be negative and (c + a) will always be negative; thus, the ratio will always result in a positive value. Therefore, this expression could be a potential solution to our problem!

Conclusion: Selecting the Correct Expression

So, after careful consideration, the correct answer is D: (a+b) / (c+a). In all the expressions, we have seen that the variables are not independent; therefore, we must consider different scenarios and test out different values to make sure which one works. In this question, we have tested multiple possible conditions and understood how the signs of the variables can affect the overall outcome. By carefully analyzing each expression and considering various possibilities for the variables, we can pinpoint the one that consistently produces a positive value. This process underscores the significance of grasping the rules of inequalities and how operations impact signs.

I hope this explanation has been helpful, guys! Always remember to break down complex problems into smaller, manageable parts. Good luck with your studies, and keep up the great work in mathematics! Remember that the more you practice these concepts, the better you'll get at solving these types of problems. Keep it up, you got this!