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]

Sure! Let's determine which expressions are equivalent to [tex]$-9\left(\frac{2}{3} x + 1\right)$[/tex] using the distributive property.

Step 1: Apply the Distributive Property

The original expression is [tex]$-9\left(\frac{2}{3} x + 1\right)$[/tex]. To apply the distributive property, we multiply [tex]$-9$[/tex] by each term inside the parentheses:

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

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

Step 2: Combine the Results

Putting those results together, the expression simplifies to:
[tex]\[
-6x - 9
\][/tex]

Step 3: Check the Given Options

Now, let's compare this with the given expressions:

1. [tex]$-9\left(\frac{2}{3} x\right)+9(1)$[/tex]: This simplifies to [tex]$-6x + 9$[/tex], which is not equivalent.
2. [tex]$-9\left(\frac{2}{3} x\right)-9(1)$[/tex]: This simplifies to [tex]$-6x - 9$[/tex], which matches our result.
3. [tex]$-9\left(\frac{2}{3} x\right)+1$[/tex]: This simplifies to [tex]$-6x + 1$[/tex], which is not equivalent.
4. [tex]$-6x + 1$[/tex]: This is not equivalent.
5. [tex]$-6x + 9$[/tex]: This is not equivalent.
6. [tex]$-6x - 9$[/tex]: This matches our result.

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]