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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] by simplifying it.

1. Simplify the Expression:

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

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

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

Calculate each part:
[tex]\[
= -6x - 9
\][/tex]

2. Compare with Given Expressions:

Let's compare [tex]\(-6x - 9\)[/tex] with the given options:

Option 1: [tex]\(-9\left(\frac{2}{3} x\right) + 9(1)\)[/tex]
- Simplify: [tex]\(-6x + 9\)[/tex]
- This does not match [tex]\(-6x - 9\)[/tex].

Option 2: [tex]\(-9\left(\frac{2}{3} x\right) - 9(1)\)[/tex]
- Simplify: [tex]\(-6x - 9\)[/tex]
- This matches our original simplified expression.

Option 3: [tex]\(-9\left(\frac{2}{3} x\right) + 1\)[/tex]
- Simplify: [tex]\(-6x + 1\)[/tex]
- This does not match [tex]\(-6x - 9\)[/tex].

Option 4: [tex]\(-6x + 1\)[/tex]
- This does not match [tex]\(-6x - 9\)[/tex].

Option 5: [tex]\(-6x + 9\)[/tex]
- This does not match [tex]\(-6x - 9\)[/tex].

Option 6: [tex]\(-6x - 9\)[/tex]
- This is exactly the same as our simplification.

3. Conclusion:

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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