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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 determine which expressions are equivalent to [tex]\(-9\left(\frac{2}{3} x + 1\right)\)[/tex] using the distributive property.

Step 1: Apply the Distributive Property

The original expression is [tex]\(-9\left(\frac{2}{3} x + 1\right)\)[/tex]. To apply the distributive property, we multiply [tex]\(-9\)[/tex] by each term inside the parentheses:

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

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

Step 2: Combine the Results

Putting those results together, the expression simplifies to:
[tex]\[
-6x - 9
\][/tex]

Step 3: Check the Given Options

Now, let's compare this with the given expressions:

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 matches our result.
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 matches our result.

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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