Answer :
We start with the expression
$$-9\left(\frac{2}{3}x + 1\right).$$
Step 1. Distribute the $-9$ to both terms inside the parentheses:
$$-9\left(\frac{2}{3}x\right) = -9 \cdot \frac{2}{3}x = -\frac{18}{3}x = -6x,$$
and
$$-9(1) = -9.$$
Step 2. Write the simplified expression:
$$-9\left(\frac{2}{3}x + 1\right) = -6x - 9.$$
Step 3. Compare the simplified expression with each given option:
1) $-9\left(\frac{2}{3}x\right) + 9(1)$ becomes $-6x + 9$, which does not match.
2) $-9\left(\frac{2}{3}x\right) - 9(1)$ becomes $-6x - 9$, which matches.
3) $-9\left(\frac{2}{3}x\right) + 1$ becomes $-6x + 1$, which does not match.
4) $-6x + 1$ does not match since we need $-6x - 9$.
5) $-6x + 9$ does not match.
6) $-6x - 9$ exactly matches the simplified expression.
Thus, the expressions that are equivalent to $-9\left(\frac{2}{3}x + 1\right)$ are in options 2 and 6.
$$-9\left(\frac{2}{3}x + 1\right).$$
Step 1. Distribute the $-9$ to both terms inside the parentheses:
$$-9\left(\frac{2}{3}x\right) = -9 \cdot \frac{2}{3}x = -\frac{18}{3}x = -6x,$$
and
$$-9(1) = -9.$$
Step 2. Write the simplified expression:
$$-9\left(\frac{2}{3}x + 1\right) = -6x - 9.$$
Step 3. Compare the simplified expression with each given option:
1) $-9\left(\frac{2}{3}x\right) + 9(1)$ becomes $-6x + 9$, which does not match.
2) $-9\left(\frac{2}{3}x\right) - 9(1)$ becomes $-6x - 9$, which matches.
3) $-9\left(\frac{2}{3}x\right) + 1$ becomes $-6x + 1$, which does not match.
4) $-6x + 1$ does not match since we need $-6x - 9$.
5) $-6x + 9$ does not match.
6) $-6x - 9$ exactly matches the simplified expression.
Thus, the expressions that are equivalent to $-9\left(\frac{2}{3}x + 1\right)$ are in options 2 and 6.
