Answer :
To determine which expressions are equivalent to [tex]\(-9\left(\frac{2}{3} x+1\right)\)[/tex], let's simplify the given expression.
1. Start with the original expression:
[tex]\(-9\left(\frac{2}{3} x + 1\right)\)[/tex]
2. Distribute the [tex]\(-9\)[/tex] across the terms inside the parentheses:
- Multiply [tex]\(-9\)[/tex] by [tex]\(\frac{2}{3} x\)[/tex]:
[tex]\[
-9 \times \frac{2}{3} x = -6x
\][/tex]
- Multiply [tex]\(-9\)[/tex] by [tex]\(1\)[/tex]:
[tex]\[
-9 \times 1 = -9
\][/tex]
3. Combine these results into a single expression:
So, the expression becomes:
[tex]\[
-6x - 9
\][/tex]
Now, compare this simplified expression to the options provided:
- [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]\(-6x + 1\)[/tex]
- [tex]\(-6x + 9\)[/tex]
- [tex]\(-6x - 9\)[/tex]
The expression [tex]\(-6x - 9\)[/tex] matches exactly with the last option given. Therefore, the equivalent expression to the original is:
- [tex]\(-6x - 9\)[/tex]
This is the only option among the list that is equivalent to the provided expression after simplification.
1. Start with the original expression:
[tex]\(-9\left(\frac{2}{3} x + 1\right)\)[/tex]
2. Distribute the [tex]\(-9\)[/tex] across the terms inside the parentheses:
- Multiply [tex]\(-9\)[/tex] by [tex]\(\frac{2}{3} x\)[/tex]:
[tex]\[
-9 \times \frac{2}{3} x = -6x
\][/tex]
- Multiply [tex]\(-9\)[/tex] by [tex]\(1\)[/tex]:
[tex]\[
-9 \times 1 = -9
\][/tex]
3. Combine these results into a single expression:
So, the expression becomes:
[tex]\[
-6x - 9
\][/tex]
Now, compare this simplified expression to the options provided:
- [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]\(-6x + 1\)[/tex]
- [tex]\(-6x + 9\)[/tex]
- [tex]\(-6x - 9\)[/tex]
The expression [tex]\(-6x - 9\)[/tex] matches exactly with the last option given. Therefore, the equivalent expression to the original is:
- [tex]\(-6x - 9\)[/tex]
This is the only option among the list that is equivalent to the provided expression after simplification.