Answer :
To determine which expressions are equivalent to [tex]\(-9\left(\frac{2}{3} x + 1\right)\)[/tex], we need to apply the distributive property to simplify the original expression.
### Step 1: Use the Distributive Property
The distributive property states that you multiply each term inside the parentheses by the factor outside. So, we distribute [tex]\(-9\)[/tex] to both terms inside the parentheses:
[tex]\[
-9\left(\frac{2}{3} x + 1\right) = -9 \cdot \frac{2}{3} x + (-9) \cdot 1
\][/tex]
### Step 2: Simplify Each Term
1. Calculate [tex]\(-9 \cdot \frac{2}{3} x\)[/tex]:
[tex]\(-9 \cdot \frac{2}{3} = \frac{-9 \cdot 2}{3} = \frac{-18}{3} = -6\)[/tex].
So, [tex]\(-9 \cdot \frac{2}{3} x = -6x\)[/tex].
2. Calculate [tex]\(-9 \cdot 1\)[/tex]:
[tex]\(-9 \cdot 1 = -9\)[/tex].
### Final Expression
Putting these results together, we have:
[tex]\[
-6x - 9
\][/tex]
### Comparison with Given Options
Now let's compare our result [tex]\(-6x - 9\)[/tex] with each of the expressions provided:
1. Option: [tex]\(-9\left(\frac{2}{3} x\right) - 9(1)\)[/tex]
- When distributed, gives [tex]\(-6x - 9\)[/tex]. This is equivalent.
2. Option: [tex]\(-9\left(\frac{2}{3} x\right) + 9(1)\)[/tex]
- Results in [tex]\(-6x + 9\)[/tex]. This is not equivalent.
3. Option: [tex]\(-6x + 9\)[/tex]
- This does not match [tex]\(-6x - 9\)[/tex]. This is not equivalent.
4. Option: [tex]\(-6x - 9\)[/tex]
- This matches exactly. This is equivalent.
5. Option: [tex]\(-6x + 1\)[/tex]
- This does not match [tex]\(-6x - 9\)[/tex]. This is not equivalent.
### Conclusion
The expressions 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]
### Step 1: Use the Distributive Property
The distributive property states that you multiply each term inside the parentheses by the factor outside. So, we distribute [tex]\(-9\)[/tex] to both terms inside the parentheses:
[tex]\[
-9\left(\frac{2}{3} x + 1\right) = -9 \cdot \frac{2}{3} x + (-9) \cdot 1
\][/tex]
### Step 2: Simplify Each Term
1. Calculate [tex]\(-9 \cdot \frac{2}{3} x\)[/tex]:
[tex]\(-9 \cdot \frac{2}{3} = \frac{-9 \cdot 2}{3} = \frac{-18}{3} = -6\)[/tex].
So, [tex]\(-9 \cdot \frac{2}{3} x = -6x\)[/tex].
2. Calculate [tex]\(-9 \cdot 1\)[/tex]:
[tex]\(-9 \cdot 1 = -9\)[/tex].
### Final Expression
Putting these results together, we have:
[tex]\[
-6x - 9
\][/tex]
### Comparison with Given Options
Now let's compare our result [tex]\(-6x - 9\)[/tex] with each of the expressions provided:
1. Option: [tex]\(-9\left(\frac{2}{3} x\right) - 9(1)\)[/tex]
- When distributed, gives [tex]\(-6x - 9\)[/tex]. This is equivalent.
2. Option: [tex]\(-9\left(\frac{2}{3} x\right) + 9(1)\)[/tex]
- Results in [tex]\(-6x + 9\)[/tex]. This is not equivalent.
3. Option: [tex]\(-6x + 9\)[/tex]
- This does not match [tex]\(-6x - 9\)[/tex]. This is not equivalent.
4. Option: [tex]\(-6x - 9\)[/tex]
- This matches exactly. This is equivalent.
5. Option: [tex]\(-6x + 1\)[/tex]
- This does not match [tex]\(-6x - 9\)[/tex]. This is not equivalent.
### Conclusion
The expressions 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]
