Which expressions are equivalent to [tex]-9\left(\frac{2}{3} x+1\right)[/tex]? Check all that apply.

A. [tex]-9\left(\frac{2}{3} x\right)+9(1)[/tex]
B. [tex]-9\left(\frac{2}{3} x\right)-9(1)[/tex]
C. [tex]-9\left(\frac{2}{3} x\right)+1[/tex]
D. [tex]-6 x+1[/tex]
E. [tex]-6 x+9[/tex]
F. [tex]-6 x-9[/tex]

Let's solve the problem step-by-step to determine which expressions are equivalent to [tex]$-9\left(\frac{2}{3} x+1\right)$[/tex].

1. Start with the Expression:

The original expression is:
[tex]\[
-9\left(\frac{2}{3} x + 1\right)
\][/tex]

2. Distribute the [tex]$-9$[/tex]:

We need to distribute [tex]$-9$[/tex] to both terms inside the parentheses:

[tex]\[
-9 \times \frac{2}{3} x + (-9) \times 1
\][/tex]

3. Calculate Each Term:

- The first term:
[tex]\[
-9 \times \frac{2}{3} x = -6x
\][/tex]
(because [tex]$-9 \times \frac{2}{3} = -6$[/tex])

- The second term:
[tex]\[
(-9) \times 1 = -9
\][/tex]

4. Combine the Terms:

Combining both terms, we get:
[tex]\[
-6x - 9
\][/tex]

Now, let's compare our result, [tex]$-6x - 9$[/tex], to the choices provided:

- [tex]$-9\left(\frac{2}{3} x\right)+9(1)$[/tex]

- [tex]$-9\left(\frac{2}{3} x\right)-9(1)$[/tex]

- [tex]$-9\left(\frac{2}{3} x\right)+1$[/tex]

- [tex]$-6x + 1$[/tex]

- [tex]$-6x + 9$[/tex]

- [tex]$-6x - 9$[/tex]

Among these options, the expressions that match [tex]$-6x - 9$[/tex] are:
- [tex]$-9\left(\frac{2}{3} x\right)-9(1)$[/tex]: This is equivalent after distributing.
- [tex]$-6x - 9$[/tex]: This is exactly the simplified form we found.

These two options are equivalent to the original expression [tex]$-9\left(\frac{2}{3} x+1\right)$[/tex].