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]

Sure, let's work through the problem step by step to find the expressions equivalent to [tex]$-9\left(\frac{2}{3}x+1\right)$[/tex].

First, let's simplify the original expression by distributing [tex]$-9$[/tex] to each term inside the parentheses:

1. Distribute [tex]$-9$[/tex] to [tex]$\frac{2}{3}x$[/tex]:
[tex]\[
-9 \times \frac{2}{3}x = -6x
\][/tex]

2. Distribute [tex]$-9$[/tex] to [tex]$1$[/tex]:
[tex]\[
-9 \times 1 = -9
\][/tex]

Combine these results to get the simplified expression:
[tex]\[
-6x - 9
\][/tex]

Now, let's compare this simplified expression to each of the given options:

1. [tex]$-9\left(\frac{2}{3} x\right) + 9(1)$[/tex]:
This simplifies to [tex]$-6x + 9$[/tex], which is not equivalent to [tex]$-6x - 9$[/tex].

2. [tex]$-9\left(\frac{2}{3} x\right) - 9(1)$[/tex]:
This simplifies to [tex]$-6x - 9$[/tex], which matches the simplified expression.

3. [tex]$-9\left(\frac{2}{3} x\right) + 1$[/tex]:
This simplifies to [tex]$-6x + 1$[/tex], which is not equivalent to [tex]$-6x - 9$[/tex].

4. [tex]$-6x + 1$[/tex]:
This expression is not equivalent to [tex]$-6x - 9$[/tex].

5. [tex]$-6x + 9$[/tex]:
This expression is not equivalent to [tex]$-6x - 9$[/tex].

6. [tex]$-6x - 9$[/tex]:
This matches the simplified expression [tex]$-6x - 9$[/tex].

So, the equivalent expressions to [tex]$-9\left(\frac{2}{3} x+1\right)$[/tex] are:
- [tex]$-9\left(\frac{2}{3} x\right)-9(1)$[/tex]
- [tex]$-6x - 9$[/tex]