High School

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 solve the problem of finding which expressions are equivalent to [tex]\(-9\left(\frac{2}{3}x + 1\right)\)[/tex], let's break it down step-by-step:

1. Distribute [tex]\(-9\)[/tex] through the parentheses:

The original 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]

2. Combine the results:

So, the expression simplifies to:
[tex]\[
-6x - 9
\][/tex]

Now, let's see which of the given options are equivalent to [tex]\(-6x - 9\)[/tex].

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

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

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

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

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

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

- Option 4: [tex]\(-6x + 1\)[/tex]

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

- Option 5: [tex]\(-6x + 9\)[/tex]

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

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

This is equivalent to the simplified expression.

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

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