Which expressions are equivalent to [tex]-9\left(\frac{2}{3} \pi+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, I can help explain why certain expressions are equivalent to the original expression [tex]$-9\left(\frac{2}{3} \pi+1\right)$[/tex].

Let's start by simplifying the original expression step-by-step:

1. Distribute the Multiplication:
[tex]\[
-9\left(\frac{2}{3} \pi + 1\right) = -9 \cdot \frac{2}{3} \pi - 9 \cdot 1
\][/tex]

2. Calculate Each Part:
- The first part is: [tex]$-9 \cdot \frac{2}{3} \pi = -6\pi$[/tex]
- The second part is: [tex]$-9 \cdot 1 = -9$[/tex]

3. Combine the Parts:
[tex]\[
-6\pi - 9
\][/tex]

Now let's analyze each given expression to see if it matches [tex]$-6x - 9$[/tex] when [tex]$x = \pi$[/tex]:

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

- [tex]$-9\left(\frac{2}{3} x\right)-9(1)$[/tex]: This evaluates to [tex]$-6x - 9$[/tex], which matches our simplified expression.

- [tex]$-9\left(\frac{2}{3} x\right)+1$[/tex]: This evaluates to [tex]$-6x + 1$[/tex], which does not match.

- [tex]$-6x+1$[/tex]: This is clearly not equal to [tex]$-6x - 9$[/tex].

- [tex]$-6x+9$[/tex]: This also does not match [tex]$-6x - 9$[/tex].

- [tex]$-6x-9$[/tex]: This expression matches exactly with our simplified expression.

Based on this breakdown, the expressions that are equivalent to [tex]$-9\left(\frac{2}{3} \pi+1\right)$[/tex] are:

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

These are the expressions that simplify to [tex]$-6x - 9$[/tex] when substituting [tex]$x = \pi$[/tex].