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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 work through the problem step by step to find the expressions equivalent to [tex]\(-9\left(\frac{2}{3}x+1\right)\)[/tex].

First, let's simplify the original expression by distributing [tex]\(-9\)[/tex] to each term inside the parentheses:

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

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

Combine these results to get the simplified expression:
[tex]\[
-6x - 9
\][/tex]

Now, let's compare this simplified expression to each of 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 to [tex]\(-6x - 9\)[/tex].

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

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

4. [tex]\(-6x + 1\)[/tex]:
This expression is not equivalent to [tex]\(-6x - 9\)[/tex].

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

6. [tex]\(-6x - 9\)[/tex]:
This matches the simplified expression [tex]\(-6x - 9\)[/tex].

So, the equivalent expressions 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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