Question

which expression is equivalent to 3(2/3p + 3 - 1/3p - 5)? choose all that apply

a. p - 6

b. p - 2

c. 3(p - 2)

d. 3(1/3p - 2)

e. 3( 1/3p + 3 - 5)

f. 3(1/3p - 8)

Answer 1

To find which expressions are equivalent to \( 3\left(\frac{2}{3}p + 3 - \frac{1}{3}p - 5\right) \), we first simplify the expression inside the parentheses.

1. Start with the expression:
\[
\frac{2}{3}p + 3 - \frac{1}{3}p - 5
\]

2. Combine like terms:
- For \( p \) terms:
\(\frac{2}{3}p - \frac{1}{3}p = \frac{1}{3}p\)
- For constant terms:
\(3 - 5 = -2\)

So we have:
\[
\frac{1}{3}p - 2
\]

3. Now multiply the entire expression by 3:
\[
3\left(\frac{1}{3}p - 2\right) = p - 6
\]

Now, we can compare this result (\( p - 6 \)) with the choices given:

- **a.** \( p - 6 \) (this is equivalent)
- **b.** \( p - 2 \) (not equivalent)
- **c.** \( 3(p - 2) = 3p - 6 \) (not equivalent)
- **d.** \( 3\left(\frac{1}{3}p - 2\right) = p - 6 \) (this is equivalent)
- **e.** \( 3\left(\frac{1}{3}p + 3 - 5\right) = 3\left(\frac{1}{3}p - 2\right) = p - 6 \) (this is equivalent)
- **f.** \( 3\left(\frac{1}{3}p - 8\right) = p - 24\) (not equivalent)

So the final list of equivalent expressions is:
- **a.** \( p - 6 \)
- **d.** \( 3\left(\frac{1}{3}p - 2\right) \)
- **e.** \( 3\left(\frac{1}{3}p + 3 - 5\right) \)

Thus, the correct answers are **a, d, and e**.