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Which expressions are equivalent to [tex]-9\left(\frac{2}{3} x+1\right)[/tex]? Check the two 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]-6 x+9[/tex]
D. [tex]-6 x-9[/tex]
E. [tex]-6 x+1[/tex]

Answer :

To determine which expressions are equivalent to [tex]\(-9\left(\frac{2}{3} x + 1\right)\)[/tex], we need to apply the distributive property to simplify the original expression.

### Step 1: Use the Distributive Property
The distributive property states that you multiply each term inside the parentheses by the factor outside. So, we distribute [tex]\(-9\)[/tex] to both terms inside the parentheses:

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

### Step 2: Simplify Each Term
1. Calculate [tex]\(-9 \cdot \frac{2}{3} x\)[/tex]:

[tex]\(-9 \cdot \frac{2}{3} = \frac{-9 \cdot 2}{3} = \frac{-18}{3} = -6\)[/tex].

So, [tex]\(-9 \cdot \frac{2}{3} x = -6x\)[/tex].

2. Calculate [tex]\(-9 \cdot 1\)[/tex]:

[tex]\(-9 \cdot 1 = -9\)[/tex].

### Final Expression
Putting these results together, we have:

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

### Comparison with Given Options
Now let's compare our result [tex]\(-6x - 9\)[/tex] with each of the expressions provided:

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

- When distributed, gives [tex]\(-6x - 9\)[/tex]. This is equivalent.

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

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

3. Option: [tex]\(-6x + 9\)[/tex]

- This does not match [tex]\(-6x - 9\)[/tex]. This is not equivalent.

4. Option: [tex]\(-6x - 9\)[/tex]

- This matches exactly. This is equivalent.

5. Option: [tex]\(-6x + 1\)[/tex]

- This does not match [tex]\(-6x - 9\)[/tex]. This is not equivalent.

### Conclusion
The expressions 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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