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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 find which expressions are equivalent to [tex]\(-9\left(\frac{2}{3} x + 1\right)\)[/tex], we need to distribute [tex]\(-9\)[/tex] to both terms inside the parentheses. Let's go through this step-by-step:

1. Distribute [tex]\(-9\)[/tex] to [tex]\(\frac{2}{3}x\)[/tex]:

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

2. Distribute [tex]\(-9\)[/tex] to [tex]\(1\)[/tex]:

[tex]\[
-9 \times 1 = -9
\][/tex]

Putting these together, the expression simplifies to:

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

Now we can compare this expression [tex]\(-6x - 9\)[/tex] with the given options:

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

2. [tex]\(-9\left(\frac{2}{3} x\right) - 9(1)\)[/tex]:\
This simplifies to [tex]\(-6x - 9\)[/tex], which is equivalent.

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

4. [tex]\(-6x + 1\)[/tex]:\
This is not equivalent.

5. [tex]\(-6x + 9\)[/tex]:\
This is not equivalent.

6. [tex]\(-6x - 9\)[/tex]:\
This is equivalent.

Therefore, 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]

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