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 perform the steps below:

1. Distribute [tex]\(-9\)[/tex] across the terms in the parentheses:

- Multiply [tex]\(-9\)[/tex] by [tex]\(\frac{2}{3}x\)[/tex]:
[tex]\[
-9 \cdot \frac{2}{3}x = -6x
\][/tex]

- Multiply [tex]\(-9\)[/tex] by [tex]\(1\)[/tex]:
[tex]\[
-9 \cdot 1 = -9
\][/tex]

2. Combine the distributed terms:

The expression becomes [tex]\(-6x - 9\)[/tex].

3. Check each given expression for equivalence to [tex]\(-6x - 9\)[/tex]:

- Expression 1: [tex]\(-9\left(\frac{2}{3} x\right) + 9(1)\)[/tex]
This simplifies to [tex]\(-6x + 9\)[/tex]. It does not match [tex]\(-6x - 9\)[/tex].

- Expression 2: [tex]\(-9\left(\frac{2}{3} x\right) - 9(1)\)[/tex]
This simplifies to [tex]\(-6x - 9\)[/tex]. It matches [tex]\(-6x - 9\)[/tex].

- Expression 3: [tex]\(-9\left(\frac{2}{3} x\right) + 1\)[/tex]
This simplifies to [tex]\(-6x + 1\)[/tex]. It does not match [tex]\(-6x - 9\)[/tex].

- Expression 4: [tex]\(-6x + 1\)[/tex]
This is the expression as given. It does not match [tex]\(-6x - 9\)[/tex].

- Expression 5: [tex]\(-6x + 9\)[/tex]
This does not match [tex]\(-6x - 9\)[/tex].

- Expression 6: [tex]\(-6x - 9\)[/tex]
This matches the simplified expression [tex]\(-6x - 9\)[/tex].

4. Conclusion:

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]