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]

Let's simplify the expression [tex]$-9\left(\frac{2}{3} x + 1\right)$[/tex] step-by-step to identify which of the given expressions are equivalent to it.

1. Distribute the [tex]$-9$[/tex] into the expression inside the parentheses:

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

2. Calculate each part of the distribution:

- [tex]$ -9 \cdot \frac{2}{3} x = -\frac{18}{3} x = -6x $[/tex]
- [tex]$(-9) \cdot 1 = -9$[/tex]

3. Combine the results:

[tex]\[
-6x - 9
\][/tex]

Now, let's compare this simplified expression [tex]$-6x - 9$[/tex] with the given choices:

- Expression 1: [tex]$-9\left(\frac{2}{3} x\right) + 9(1)$[/tex]
- Simplifies to: [tex]$-6x + 9$[/tex]
- Not equivalent

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

- Expression 3: [tex]$-9\left(\frac{2}{3} x\right) + 1$[/tex]
- Simplifies to: [tex]$-6x + 1$[/tex]
- Not equivalent

- Expression 4: [tex]$-6x + 1$[/tex]
- Not equivalent

- Expression 5: [tex]$-6x + 9$[/tex]
- Not equivalent

- Expression 6: [tex]$-6x - 9$[/tex]
- Equivalent

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

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

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]

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