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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]

Answer :

To determine which expressions are equivalent to [tex]\(-9\left(\frac{2}{3} x+1\right)\)[/tex], let's simplify the given expression.

1. Start with the original expression:

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

2. Distribute the [tex]\(-9\)[/tex] across the terms inside the parentheses:

- Multiply [tex]\(-9\)[/tex] by [tex]\(\frac{2}{3} x\)[/tex]:
[tex]\[
-9 \times \frac{2}{3} x = -6x
\][/tex]

- Multiply [tex]\(-9\)[/tex] by [tex]\(1\)[/tex]:
[tex]\[
-9 \times 1 = -9
\][/tex]

3. Combine these results into a single expression:

So, the expression becomes:
[tex]\[
-6x - 9
\][/tex]

Now, compare this simplified expression to the options provided:

- [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]\(-6x + 1\)[/tex]
- [tex]\(-6x + 9\)[/tex]
- [tex]\(-6x - 9\)[/tex]

The expression [tex]\(-6x - 9\)[/tex] matches exactly with the last option given. Therefore, the equivalent expression to the original is:

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

This is the only option among the list that is equivalent to the provided expression after simplification.

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