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]

To determine which expressions are equivalent to [tex]$-9\left(\frac{2}{3} x + 1\right)$[/tex], we need to simplify and distribute the expression step-by-step.

1. Start with the original expression:
[tex]\[
-9\left(\frac{2}{3} x + 1\right)
\][/tex]

2. Distribute the [tex]$-9$[/tex] inside the parentheses:
- Multiply [tex]$-9$[/tex] by [tex]$\frac{2}{3} x$[/tex]:
[tex]\[
-9 \times \frac{2}{3} x = -6x
\][/tex]

- Multiply [tex]$-9$[/tex] by [tex]$1$[/tex]:
[tex]\[
-9 \times 1 = -9
\][/tex]

3. Combine the results:
[tex]\[
-6x - 9
\][/tex]

Now, let's check which of the given options are equivalent to [tex]$-6x - 9$[/tex]:

- Option 1: [tex]$-9\left(\frac{2}{3} x\right) + 9(1)$[/tex]
- Simplifies to: [tex]$-6x + 9$[/tex]. Not equivalent because the constant term is [tex]$+9$[/tex] instead of [tex]$-9$[/tex].

- Option 2: [tex]$-9\left(\frac{2}{3} x\right) - 9(1)$[/tex]
- Simplifies to: [tex]$-6x - 9$[/tex]. Equivalent.

- Option 3: [tex]$-9\left(\frac{2}{3} x\right) + 1$[/tex]
- Simplifies to: [tex]$-6x + 1$[/tex]. Not equivalent because the constant term is [tex]$+1$[/tex] instead of [tex]$-9$[/tex].

- Option 4: [tex]$-6x + 1$[/tex]
- This is not equivalent because the constant term is [tex]$+1$[/tex] instead of [tex]$-9$[/tex].

- Option 5: [tex]$-6x + 9$[/tex]
- This is not equivalent because the constant term is [tex]$+9$[/tex] instead of [tex]$-9$[/tex].

- Option 6: [tex]$-6x - 9$[/tex]
- This is exactly our simplified expression. Equivalent.

Therefore, the expressions that are equivalent 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]