College

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 find which expressions are equivalent to [tex]\(-9\left(\frac{2}{3} x + 1\right)\)[/tex], we need to distribute the [tex]\(-9\)[/tex] across each term inside the parentheses. Let's do this step-by-step:

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

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

2. Calculate each part:

- [tex]\(-9 \times \frac{2}{3} x = -6x\)[/tex]
(because [tex]\(-9 \times \frac{2}{3} = -6\)[/tex])

- [tex]\(-9 \times 1 = -9\)[/tex]

3. Combine the results:

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

Now, this is the expression we came up with: [tex]\(-6x - 9\)[/tex].

Let's check each option to see which ones match:

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

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

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

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

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

6. [tex]\(-6x - 9\)[/tex]:
This is exactly what we found, so it is equivalent to [tex]\(-6x - 9\)[/tex].

So, 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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