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 find which expressions are equivalent to [tex]\(-9\left(\frac{2}{3} x+1\right)\)[/tex], we'll start by simplifying the given expression by distributing the [tex]\(-9\)[/tex] across the terms inside the parentheses.

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

2. Distribute -9 to each term 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, we'll check the provided expressions to see which ones are equivalent to [tex]\(-6x - 9\)[/tex].

- First option: [tex]\(-9\left(\frac{2}{3} x\right) + 9(1)\)[/tex]
- Simplifies to: [tex]\(-6x + 9\)[/tex]

- Second option: [tex]\(-9\left(\frac{2}{3} x\right) - 9(1)\)[/tex]
- Simplifies to: [tex]\(-6x - 9\)[/tex]
- This matches our simplified expression.

- Third option: [tex]\(-9\left(\frac{2}{3} x\right) + 1\)[/tex]
- Simplifies to: [tex]\(-6x + 1\)[/tex]

- Fourth option: [tex]\(-6x + 1\)[/tex]
- This is already in simplified form: [tex]\(-6x + 1\)[/tex]

- Fifth option: [tex]\(-6x + 9\)[/tex]
- This is already in simplified form: [tex]\(-6x + 9\)[/tex]

- Sixth option: [tex]\(-6x - 9\)[/tex]
- This is already in simplified form and matches our expression: [tex]\(-6x - 9\)[/tex]

Based on our analysis, 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]