Which expressions are equivalent to [tex]-9\left(\frac{2}{3} x+1\right)[/tex]? Check all that apply.

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

To solve this problem, we want to determine which expressions are equivalent to the expression [tex]$-9\left(\frac{2}{3} x+1\right)$[/tex]. Let's break it down step-by-step:

1. Distribute the -9 in the expression:

We have [tex]$-9\left(\frac{2}{3} x+1\right)$[/tex]. We need to distribute the [tex]$-9$[/tex] to each term inside the parentheses:

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

- Next, distribute [tex]$-9$[/tex] to [tex]$1$[/tex]:
[tex]\[
-9 \times 1 = -9
\][/tex]

So, the expression simplifies to [tex]$-6x - 9$[/tex].

2. Identify equivalent expressions:

We need to check which of the given expressions are equivalent to [tex]$-6x - 9$[/tex]:

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

3. Conclude which expressions are equivalent:

Comparing each of the expressions above with [tex]$-6x - 9$[/tex], we find that:

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

Therefore, the expressions [tex]$-9\left(\frac{2}{3} x\right)-9(1)$[/tex] and [tex]$-6x - 9$[/tex] are equivalent to [tex]$-9\left(\frac{2}{3} x+1\right)$[/tex].