Equivalent Expressions When X Equals 3 - Solving For -4x-8

Equivalent expressions in mathematics are expressions that, while potentially appearing different, yield the same value for any given value of the variable. To determine if two expressions are equivalent, one common method is to substitute a specific value for the variable and evaluate both expressions. If the resulting values are the same, it provides evidence that the expressions might be equivalent. However, it's crucial to understand that testing with a single value doesn't definitively prove equivalence; it simply provides a strong indication. To definitively prove equivalence, one must use algebraic manipulation to show that one expression can be transformed into the other.

In this scenario, Giovanni is attempting to justify whether the expression $-4x - 8$ is equivalent to the expression $-2(x + 1) - 2(x + 3)$. His approach involves substituting $x = 3$ into both expressions and comparing the results. This method aligns with the initial step in verifying equivalence, but it's essential to remember that further steps might be necessary for a conclusive proof. The core idea is that if two expressions are truly equivalent, they should produce the same output for any input value. By choosing $x = 3$, Giovanni is testing this hypothesis for a specific case. If the values differ, it immediately demonstrates that the expressions are not equivalent. If the values are the same, it suggests equivalence, but further validation is recommended.

It's also important to note that while substituting a single value can disprove equivalence, it cannot definitively prove it. For instance, if both expressions evaluate to the same value for $x = 3$, it doesn't guarantee they will do so for all values of $x$. There might be specific values of $x$ for which the expressions diverge. To rigorously prove equivalence, one typically employs algebraic techniques such as distribution, combining like terms, and factoring. These techniques allow for the transformation of one expression into the other, thereby demonstrating their equivalence for all possible values of the variable.

To determine the value of each expression when $x = 3$, we will substitute this value into both expressions and perform the necessary arithmetic operations. This process involves replacing the variable $x$ with the numerical value 3 and then simplifying the expression according to the order of operations (PEMDAS/BODMAS). This method allows us to directly compare the results of the two expressions for the specific case of $x = 3$. If the results are identical, it supports the hypothesis that the expressions might be equivalent. However, if the results differ, it immediately demonstrates that the expressions are not equivalent for all values of $x$.

First, let's evaluate the expression $-4x - 8$ when $x = 3$. We substitute $x$ with 3, resulting in $-4(3) - 8$. Following the order of operations, we perform the multiplication first: $-4 imes 3 = -12$. Then, we subtract 8 from -12: $-12 - 8 = -20$. Therefore, the value of the expression $-4x - 8$ when $x = 3$ is -20. This result provides a specific numerical value that we can compare against the value of the second expression when $x = 3$.

Next, let's evaluate the expression $-2(x + 1) - 2(x + 3)$ when $x = 3$. We substitute $x$ with 3, resulting in $-2(3 + 1) - 2(3 + 3)$. Following the order of operations, we first evaluate the expressions inside the parentheses: $3 + 1 = 4$ and $3 + 3 = 6$. Now we have $-2(4) - 2(6)$. Next, we perform the multiplications: $-2 imes 4 = -8$ and $-2 imes 6 = -12$. Finally, we subtract -12 from -8: $-8 - 12 = -20$. Therefore, the value of the expression $-2(x + 1) - 2(x + 3)$ when $x = 3$ is also -20. This result matches the value of the first expression, suggesting that the expressions might be equivalent, but further verification is needed for a conclusive proof.

After evaluating both expressions at $x = 3$, we obtained the value -20 for both $-4x - 8$ and $-2(x + 1) - 2(x + 3)$. This outcome indicates that when $x$ is equal to 3, the two expressions yield the same result. However, it's crucial to understand that this single instance of agreement does not definitively prove that the two expressions are equivalent for all possible values of $x$. It merely provides a strong indication that they might be equivalent. To establish equivalence with certainty, a more rigorous approach involving algebraic manipulation is required.

Algebraic manipulation involves transforming one expression into the other through a series of valid operations, such as distribution, combining like terms, and factoring. If we can successfully transform $-2(x + 1) - 2(x + 3)$ into $-4x - 8$, or vice versa, then we can definitively conclude that the two expressions are equivalent. This is because algebraic manipulation preserves the value of the expression, ensuring that the transformed expression will always yield the same result as the original expression for any value of $x$.

In this specific case, let's attempt to simplify the expression $-2(x + 1) - 2(x + 3)$. We begin by distributing the -2 across the terms inside each set of parentheses: $-2 * x + (-2) * 1 - 2 * x + (-2) * 3$, which simplifies to $-2x - 2 - 2x - 6$. Next, we combine like terms: $-2x - 2x$ gives us $-4x$, and $-2 - 6$ gives us $-8$. Combining these results, we obtain $-4x - 8$, which is precisely the first expression. This algebraic manipulation demonstrates that the expression $-2(x + 1) - 2(x + 3)$ can be transformed into $-4x - 8$, thus proving their equivalence.

Based on our calculations, we found that both expressions, $-4x - 8$ and $-2(x + 1) - 2(x + 3)$, evaluate to -20 when $x = 3$. Therefore, the correct answer must indicate that both expressions have a value of -20. This aligns with the concept of equivalent expressions, which, if truly equivalent, should yield the same value for any given input. However, remember that verifying for a single value only provides an indication of potential equivalence; a conclusive proof requires algebraic manipulation or other rigorous methods.

Now, let's examine the provided options:

• A. 3 and -20
• B. 4 and 4
• C. -20 and -20
• D. 3 and 4

Option C, -20 and -20, is the only option that accurately reflects the values we calculated for both expressions when $x = 3$. The other options present differing values or values that do not match our calculations. Therefore, based on our evaluation, option C is the correct answer.

It's important to highlight again that while substituting a value can help identify potential equivalence or non-equivalence, it doesn't provide a definitive proof of equivalence. In this case, the substitution of $x = 3$ led us to the same value for both expressions, which aligns with the fact that the expressions are indeed equivalent. However, to be completely certain, we would still need to perform algebraic manipulation, as we demonstrated earlier, to confirm that one expression can be transformed into the other.

In conclusion, Giovanni's method of substituting $x = 3$ into both expressions is a valid initial step in checking for equivalence. Our calculations showed that both expressions, $-4x - 8$ and $-2(x + 1) - 2(x + 3)$, evaluate to -20 when $x = 3$. This result aligns with option C, making it the correct answer. However, it's essential to remember that while this substitution suggests equivalence, a definitive proof requires algebraic manipulation. We further demonstrated the equivalence by simplifying $-2(x + 1) - 2(x + 3)$ to $-4x - 8$, confirming that the two expressions are indeed equivalent for all values of $x$.

This exercise highlights the importance of understanding the concept of equivalent expressions and the methods used to verify them. While substitution can be a useful tool, algebraic manipulation provides a more rigorous and conclusive approach to proving equivalence. By combining these methods, we can confidently determine whether two expressions are equivalent and simplify mathematical problems effectively.