Identifying Equivalent Expressions

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 identify which expressions are equivalent to [tex]$-9\left(\frac{2}{3} x+1\right)$[/tex], let's start by simplifying this expression step-by-step:

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

- First, multiply [tex]$-9$[/tex] by [tex]$\frac{2}{3} x$[/tex]:
[tex]\[
-9 \times \frac{2}{3} x = -6x
\][/tex]

- Next, multiply [tex]$-9$[/tex] by [tex]$1$[/tex]:
[tex]\[
-9 \times 1 = -9
\][/tex]

So the simplified expression is:
[tex]\[
-6x - 9
\][/tex]

2. Analyze each provided option to see which are equivalent:

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

- Option 2: [tex]$-9\left(\frac{2}{3} x\right) - 9$[/tex]
- This simplifies to [tex]$-6x - 9$[/tex], which is indeed equivalent to the simplified expression.

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

- Option 4: [tex]$-6x + 1$[/tex]
- This is not equivalent to [tex]$-6x - 9$[/tex].

- Option 5: [tex]$-6x + 9$[/tex]
- This is not equivalent to [tex]$-6x - 9$[/tex].

- Option 6: [tex]$-6x - 9$[/tex]
- This directly matches the simplified expression and is equivalent.

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

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