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 find which expressions are equivalent to [tex]\(-9\left(\frac{2}{3} x+1\right)\)[/tex] by simplifying the expression. We will distribute the [tex]\(-9\)[/tex] to both terms inside the parentheses:

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

2. Distribute the [tex]\(-9\)[/tex] to each term inside the parentheses:
- First term: [tex]\(-9 \times \frac{2}{3}x = -6x\)[/tex]
- Second term: [tex]\(-9 \times 1 = -9\)[/tex]

3. Putting it all together, the expression simplifies to:
[tex]\(-6x - 9\)[/tex].

Now, let's check the given options to see which ones are equivalent to [tex]\(-6x - 9\)[/tex]:

1. [tex]\(-9\left(\frac{2}{3} x\right) + 9(1)\)[/tex]
- Simplifies to: [tex]\(-6x + 9\)[/tex]
- Not equivalent.

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

3. [tex]\(-9\left(\frac{2}{3} x\right) + 1\)[/tex]
- Simplifies to: [tex]\(-6x + 1\)[/tex]
- Not equivalent.

4. [tex]\(-6x + 1\)[/tex]
- Already simplified; not equivalent to [tex]\(-6x - 9\)[/tex].

5. [tex]\(-6x + 9\)[/tex]
- Already simplified; not equivalent to [tex]\(-6x - 9\)[/tex].

6. [tex]\(-6x - 9\)[/tex]
- Already simplified; equivalent.

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

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