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------------------------------------------------ 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 the expression by distributing the [tex]\(-9\)[/tex] across the terms inside the parentheses. Here’s how we do it step-by-step:

1. Distribution of [tex]\(-9\)[/tex]:

We have [tex]\(-9\left(\frac{2}{3}x + 1\right)\)[/tex]. We need to distribute [tex]\(-9\)[/tex] to both [tex]\(\frac{2}{3}x\)[/tex] and 1:

- 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 1:
[tex]\[
-9 \times 1 = -9
\][/tex]

2. Combine the results:

Combine the distributed terms:
[tex]\[
-6x - 9
\][/tex]

Now let's compare this with the options given to see which are equivalent:

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