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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 transform this expression step-by-step:

1. Distribute [tex]\(-9\)[/tex] into the expression:

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

2. Calculate each term separately:

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

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

3. Combine the results:

The expanded expression is:
[tex]\[
-6x - 9
\][/tex]

With the expanded expression [tex]\(-6x - 9\)[/tex], let's check each of the given options to see if they are equivalent:

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

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

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

- Option 4: [tex]\(-6x + 1\)[/tex]
Not equivalent since [tex]\(-6x + 1 \neq -6x - 9\)[/tex].

- Option 5: [tex]\(-6x + 9\)[/tex]
Not equivalent since [tex]\(-6x + 9 \neq -6x - 9\)[/tex].

- Option 6: [tex]\(-6x - 9\)[/tex]
This is exactly the same expression as our expanded form.

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

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