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Answer :
Sure! Let's work through the problem step by step to find which expressions are equivalent to [tex]\(-\frac{3}{4}(32 + 24e - 4f)\)[/tex].
First, we need to distribute [tex]\(-\frac{3}{4}\)[/tex] across each term inside the parentheses:
1. Compute [tex]\(-\frac{3}{4} \times 32\)[/tex]:
[tex]\[
-\frac{3}{4} \times 32 = -24
\][/tex]
2. Compute [tex]\(-\frac{3}{4} \times 24e\)[/tex]:
[tex]\[
-\frac{3}{4} \times 24e = -18e
\][/tex]
3. Compute [tex]\(-\frac{3}{4} \times (-4f)\)[/tex]:
[tex]\[
-\frac{3}{4} \times (-4f) = 3f
\][/tex]
Once distributed, the expression becomes [tex]\(-24 - 18e + 3f\)[/tex].
Now, let's compare this expression to each option:
- A. [tex]\(24 + 18e - 3f\)[/tex]
- This does not match [tex]\(-24 - 18e + 3f\)[/tex].
- B. [tex]\(-18e + 3f - 24\)[/tex]
- This expression is equivalent, as it has terms rearranged but is effectively the same: [tex]\(-24 - 18e + 3f\)[/tex].
- C. [tex]\(-24 - 18e + 3f\)[/tex]
- This exactly matches the distributed expression.
- D. [tex]\(8 - 6e - f\)[/tex]
- This does not match [tex]\(-24 - 18e + 3f\)[/tex].
- E. [tex]\(3(-8 - 6e + f)\)[/tex]
- This is equivalent since distributing 3 gives [tex]\(-24 - 18e + 3f\)[/tex].
Therefore, the expressions that are equivalent to [tex]\(-\frac{3}{4}(32 + 24e - 4f)\)[/tex] are B. [tex]\(-18e + 3f - 24\)[/tex], C. [tex]\(-24 - 18e + 3f\)[/tex], and E. [tex]\(3(-8 - 6e + f)\)[/tex].
First, we need to distribute [tex]\(-\frac{3}{4}\)[/tex] across each term inside the parentheses:
1. Compute [tex]\(-\frac{3}{4} \times 32\)[/tex]:
[tex]\[
-\frac{3}{4} \times 32 = -24
\][/tex]
2. Compute [tex]\(-\frac{3}{4} \times 24e\)[/tex]:
[tex]\[
-\frac{3}{4} \times 24e = -18e
\][/tex]
3. Compute [tex]\(-\frac{3}{4} \times (-4f)\)[/tex]:
[tex]\[
-\frac{3}{4} \times (-4f) = 3f
\][/tex]
Once distributed, the expression becomes [tex]\(-24 - 18e + 3f\)[/tex].
Now, let's compare this expression to each option:
- A. [tex]\(24 + 18e - 3f\)[/tex]
- This does not match [tex]\(-24 - 18e + 3f\)[/tex].
- B. [tex]\(-18e + 3f - 24\)[/tex]
- This expression is equivalent, as it has terms rearranged but is effectively the same: [tex]\(-24 - 18e + 3f\)[/tex].
- C. [tex]\(-24 - 18e + 3f\)[/tex]
- This exactly matches the distributed expression.
- D. [tex]\(8 - 6e - f\)[/tex]
- This does not match [tex]\(-24 - 18e + 3f\)[/tex].
- E. [tex]\(3(-8 - 6e + f)\)[/tex]
- This is equivalent since distributing 3 gives [tex]\(-24 - 18e + 3f\)[/tex].
Therefore, the expressions that are equivalent to [tex]\(-\frac{3}{4}(32 + 24e - 4f)\)[/tex] are B. [tex]\(-18e + 3f - 24\)[/tex], C. [tex]\(-24 - 18e + 3f\)[/tex], and E. [tex]\(3(-8 - 6e + f)\)[/tex].
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