Finding Expressions Equal To 0 When X = -1

Hey math enthusiasts! Let's dive into some cool algebra. We're on a mission to figure out which of the given expressions magically turn into 0 when we plug in x = -1. This is a classic problem that tests your understanding of algebraic expressions and how they behave with specific values. No sweat, we'll break it down step by step to make sure everyone understands. We'll be using substitution and simplification, and don't worry, it's not as scary as it sounds. We'll go through each option, replacing x with -1 and simplifying to see if we hit that sweet spot of 0. Ready to get started? Let's go!

Understanding the Problem: Expressions and Substitution

So, what's the deal, guys? We've got a bunch of algebraic expressions, and our task is to evaluate them at a specific value of x. The key concept here is substitution. Basically, we're swapping out the variable x in each expression with the value -1. Think of it like a simple trade: x is gone, and -1 takes its place. After the substitution, we'll simplify the expression using the order of operations (PEMDAS/BODMAS) to see if the result is 0. If it is, then that expression is equal to 0 when x = -1. If not, then we move on to the next expression. It's like a treasure hunt, and we're looking for the expressions that lead us to the treasure (which, in this case, is 0).

This exercise reinforces the fundamental principles of algebra. It helps in building a solid foundation in simplifying algebraic expressions. This skill is critical for more advanced math concepts. Let's get our hands dirty and start with option A!

Evaluating Each Expression

A. $\frac{4(x+1)}{(4 x+5)}$

Alright, let's start with option A. Here's our expression: $\frac{4(x+1)}{(4 x+5)}$. We need to substitute x with -1. So, the expression becomes: $\frac{4((-1)+1)}{(4 (-1)+5)}$. Now, let's simplify. Inside the parentheses in the numerator, we have (-1 + 1), which equals 0. So, the numerator becomes 4(0) = 0. The denominator is (4 * -1) + 5 = -4 + 5 = 1. Therefore, the entire expression simplifies to $\frac{0}{1}$, which equals 0. So, option A is a winner! This means that when x is -1, the expression in option A equals 0. Awesome, right? Let's keep the momentum going!

B. $\frac{4(x-1)}{(5-4 x)}$

Now, let's move onto option B. We have $\frac{4(x-1)}{(5-4 x)}$. Again, we'll substitute x with -1. This time, our expression turns into $\frac{4((-1)-1)}{(5-4 (-1))}$. Let's simplify. The numerator is 4(-1 - 1) = 4(-2) = -8. The denominator is 5 - (4 * -1) = 5 + 4 = 9. So, the simplified expression is $\frac{-8}{9}$. This is not equal to 0. So, option B is not a match. This expression does not equal 0 when x = -1. Easy peasy, right?

C. $\frac{4(x-(-1))}{(4 x+5)}$

Onwards to option C! Here, we have $\frac{4(x-(-1))}{(4 x+5)}$. Let's do the usual substitution of x with -1. The expression becomes $\frac{4((-1)-(-1))}{(4 (-1)+5)}$. Simplifying the numerator, we get (-1 - (-1)) = (-1 + 1) = 0. So, the numerator is 4 * 0 = 0. The denominator, as we saw in option A, is (4 * -1) + 5 = -4 + 5 = 1. Therefore, we have $\frac{0}{1}$, which is equal to 0. Hey, option C is also a winner! The expression in C equals 0 when x = -1.

D. $\frac{4(x+(-1))}{(4 x+5)}$

Let's check out option D. Our expression is $\frac{4(x+(-1))}{(4 x+5)}$. Substitute x with -1, resulting in $\frac{4((-1)+(-1))}{(4 (-1)+5)}$. Now, let's simplify. The numerator is 4(-1 - 1) = 4(-2) = -8. The denominator is (4 * -1) + 5 = -4 + 5 = 1. Thus, the expression becomes $\frac{-8}{1}$, which equals -8. This is definitely not equal to 0. So, option D is a no-go.

E. $\frac{4(x+1)}{(5-4 x)}$

Finally, let's tackle option E. Here we have $\frac{4(x+1)}{(5-4 x)}$. Substituting x with -1, we get $\frac{4((-1)+1)}{(5-4 (-1))}$. Let's simplify. The numerator is 4(-1 + 1) = 4 * 0 = 0. The denominator is 5 - (4 * -1) = 5 + 4 = 9. So, the expression simplifies to $\frac{0}{9}$, which equals 0. Awesome, option E is a winner!

Conclusion: The Expressions that Equal 0

Alright, guys, we've gone through each expression. Now, let's tally up our results. Remember, we were looking for expressions that equal 0 when x = -1. We found that the following expressions meet this criteria:

• A. $\frac{4(x+1)}{(4 x+5)}$
• C. $\frac{4(x-(-1))}{(4 x+5)}$
• E. $\frac{4(x+1)}{(5-4 x)}$

Congratulations! You've successfully navigated the world of algebraic expressions and substitution. This kind of problem is crucial for building your skills in algebra and other areas of mathematics. Keep practicing, and you'll become a pro in no time! Remember to always break down problems into smaller steps and double-check your work. You've got this!

Further Exploration: Beyond the Basics

Now that we've solved the problem, let's consider some extra stuff to deepen your understanding. This problem can be extended by asking other questions that relate to the same concepts. You could also find the values of x that make the expression have an undefined answer by looking into the value that makes the denominator equal to 0. You can also explore different algebraic manipulations to simplify the expressions further. For example, if we look back at the original problem. We know A and E are correct, and B, and D are not. But it would be easier if you can spot that option C is similar to option A. Therefore, when you are taking the test, and you know the answer is option A, then option C is correct too. Option D is similar to A, C and E, but the expressions are not equal to 0 when x = -1. By understanding these concepts, you're not just solving a problem; you're developing critical thinking skills applicable across various fields. Keep experimenting, keep learning, and keep the mathematical spirit alive!

Understanding these basic concepts allows you to approach more complex mathematical problems with confidence. The ability to manipulate and evaluate algebraic expressions is a foundational skill for success in algebra, calculus, and beyond. So, keep up the great work, guys! You're building a solid base for future math adventures!