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Which expressions are equivalent to [tex]-9\left(\frac{2}{3} x+1\right)[/tex]? Check all that apply.

- [tex]-9\left(\frac{2}{3} x\right)+9(1)[/tex]
- [tex]-9\left(\frac{2}{3} x\right)-9(1)[/tex]
- [tex]-9\left(\frac{2}{3} x\right)+1[/tex]
- [tex]-6 x+1[/tex]
- [tex]-6 x+9[/tex]
- [tex]-6 x-9[/tex]

Answer :

To determine which expressions are equivalent to [tex]\(-9\left(\frac{2}{3} x + 1\right)\)[/tex], we need to simplify the original expression and compare it with the options provided.

### Step 1: Simplify the Original Expression

The expression is [tex]\(-9\left(\frac{2}{3} x + 1\right)\)[/tex]. Let's distribute the [tex]\(-9\)[/tex] to both terms inside the parentheses:

1. First term: [tex]\(-9 \times \frac{2}{3} x = -\frac{18}{3} x = -6x\)[/tex]
2. Second term: [tex]\(-9 \times 1 = -9\)[/tex]

So, the expression simplifies to [tex]\(-6x - 9\)[/tex].

### Step 2: Compare with the Given Options

Let's check each option to see if they match [tex]\(-6x - 9\)[/tex]:

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 is exactly equivalent to the simplified expression [tex]\(-6x - 9\)[/tex].

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

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

6. [tex]\(-6x - 9\)[/tex]
- This matches the simplified version of the original expression, [tex]\(-6x - 9\)[/tex].

### 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]

These two options correctly represent the original expression when simplified.

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