Question

which expressions are equivalent to $-\frac{1}{4}(12 - 8 + 20x)$?
select all that apply.
a. $5x - 1$
b. $3 + 2 - 5x$
c. $5x + 1$
d. $4x$
e. $-3 + 2 - 5x$
f. $-5x - 1$

Explanation:

Step1: Simplify the given expression

First, simplify the expression inside the parentheses: \(12 - 8 + 20x = 4 + 20x\). Then apply the distributive property: \(-\frac{1}{4}(4 + 20x)=-\frac{1}{4}\times4-\frac{1}{4}\times20x=-1 - 5x\). We can also rearrange the terms: \(-5x - 1\) or \( - 3+2 - 5x\) (since \(-1=-3 + 2\)).

Step2: Analyze each option

• Option A: \(5x-1\) has a positive \(5x\) term, while our simplified expression has \(-5x\), so not equivalent.
• Option B: \(3 + 2-5x=5 - 5x\), which is not equal to \(-5x - 1\), so not equivalent. Wait, no, wait our simplified expression is \(-1-5x=-5x - 1\) and \(-3 + 2-5x=-1-5x\) (since \(-3 + 2=-1\)). Wait, let's re - check the simplification:

Wait, original expression: \(-\frac{1}{4}(12-8 + 20x)\)
First, \(12-8 = 4\), so the expression inside is \(4 + 20x\)
Then \(-\frac{1}{4}\times4= - 1\) and \(-\frac{1}{4}\times20x=-5x\), so the simplified expression is \(-5x-1\) or \(-3 + 2-5x\) (because \(-1=-3 + 2\)) and also \(3 + 2-5x\) is \(5-5x\) which is wrong. Wait, I made a mistake. Let's recalculate:
\(12-8=4\), so \(-\frac{1}{4}(4 + 20x)=- \frac{1}{4}\times4-\frac{1}{4}\times20x=-1-5x\)
Now let's check each option:

• Option A: \(5x - 1\): The coefficient of \(x\) is \(5\) instead of \(-5\), not equivalent.
• Option B: \(3+2 - 5x=5 - 5x

eq-5x - 1\), not equivalent.

• Option C: \(5x + 1\): Coefficient of \(x\) is positive and constant term is positive, not equivalent.
• Option D: \(4x\): Not equivalent.
• Option E: \(-3 + 2-5x=-1-5x\) (since \(-3 + 2=-1\)), which is equal to \(-5x - 1\), so equivalent.
• Option F: \(-5x-1\): This is exactly our simplified expression, so equivalent.

Wait, I think I messed up the first calculation of option B. Let's re - do the simplification of the original expression:

Original expression: \(-\frac{1}{4}(12-8 + 20x)\)

First, compute \(12-8 = 4\), so the expression becomes \(-\frac{1}{4}(4 + 20x)\)

Using the distributive property \(a(b + c)=ab+ac\), here \(a =-\frac{1}{4}\), \(b = 4\), \(c = 20x\)

So \(-\frac{1}{4}\times4=-\frac{4}{4}=-1\) and \(-\frac{1}{4}\times20x=-\frac{20x}{4}=-5x\)

So the simplified expression is \(-1-5x=-5x - 1\)

Now let's check each option:

• Option A: \(5x-1\): The \(x\) term is positive, our expression has \(-5x\), so not equivalent.
• Option B: \(3 + 2-5x=5-5x\): \(5-5x\) is not equal to \(-5x - 1\) (since \(5

eq - 1\)), so not equivalent.

• Option C: \(5x + 1\): \(x\) term is positive and constant term is positive, not equivalent.
• Option D: \(4x\): Not equivalent.
• Option E: \(-3+2 - 5x=-1-5x\) (because \(-3 + 2=-1\)) which is equal to \(-5x - 1\), so equivalent.
• Option F: \(-5x-1\): This is the simplified form, so equivalent.

Wait, also let's check another way. Let's rewrite \(-5x - 1\) as \(-3+2 - 5x\) (since \(-1=-3 + 2\)) and \(-5x-1\) is the same as option F. Also, is there any other option? Wait, maybe I made a mistake in the initial simplification. Wait, the original expression is \(-\frac{1}{4}(12 - 8+20x)\). Wait, \(12-8 = 4\), so \(-\frac{1}{4}(4 + 20x)=-1-5x\). Now, let's check option E: \(-3 + 2-5x=-1-5x\) (correct, because \(-3+2=-1\)), option F: \(-5x - 1\) (correct). Are there any other options? Wait, maybe I missed. Wait, let's check option B: \(3 + 2-5x=5-5x\), which is not equal to \(-5x - 1\). Option A: \(5x-1\) is not equal. Option C: \(5x + 1\) no. Option D: \(4x\) no. So the equivalent expressions are E and F. Wait, but also let's check if \(-3 + 2-5x\) is equal to \(-5x - 1\): \(-3+2=-1\), so \(-3 + 2-5x=-5x - 1\), yes. And option F is \(-5x - 1\), yes. So the correct options are E an…

Answer:

E. \(-3 + 2-5x\), F. \(-5x-1\)