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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 :

Sure! Let's solve the problem step by step without referring to any code.

We need to find expressions equivalent to [tex]\(-9\left(\frac{2}{3} x+1\right)\)[/tex].

### Step 1: Distribute [tex]\(-9\)[/tex] to each term inside the parentheses.

The expression is [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]

### Step 2: Combine the results.

Combine the two results to rewrite the expression:
[tex]\[
-6x - 9
\][/tex]

### Step 3: Check which given options are equivalent to [tex]\(-6x - 9\)[/tex].

Let's check:

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