Answer :
Let's solve the problem step-by-step to determine which expressions are equivalent to [tex]\(-9\left(\frac{2}{3} x+1\right)\)[/tex].
1. Start with the Expression:
The original expression is:
[tex]\[
-9\left(\frac{2}{3} x + 1\right)
\][/tex]
2. Distribute the [tex]\(-9\)[/tex]:
We need to distribute [tex]\(-9\)[/tex] to both terms inside the parentheses:
[tex]\[
-9 \times \frac{2}{3} x + (-9) \times 1
\][/tex]
3. Calculate Each Term:
- The first term:
[tex]\[
-9 \times \frac{2}{3} x = -6x
\][/tex]
(because [tex]\(-9 \times \frac{2}{3} = -6\)[/tex])
- The second term:
[tex]\[
(-9) \times 1 = -9
\][/tex]
4. Combine the Terms:
Combining both terms, we get:
[tex]\[
-6x - 9
\][/tex]
Now, let's compare our result, [tex]\(-6x - 9\)[/tex], to the choices 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]
Among these options, the expressions that match [tex]\(-6x - 9\)[/tex] are:
- [tex]\(-9\left(\frac{2}{3} x\right)-9(1)\)[/tex]: This is equivalent after distributing.
- [tex]\(-6x - 9\)[/tex]: This is exactly the simplified form we found.
These two options are equivalent to the original expression [tex]\(-9\left(\frac{2}{3} x+1\right)\)[/tex].
1. Start with the Expression:
The original expression is:
[tex]\[
-9\left(\frac{2}{3} x + 1\right)
\][/tex]
2. Distribute the [tex]\(-9\)[/tex]:
We need to distribute [tex]\(-9\)[/tex] to both terms inside the parentheses:
[tex]\[
-9 \times \frac{2}{3} x + (-9) \times 1
\][/tex]
3. Calculate Each Term:
- The first term:
[tex]\[
-9 \times \frac{2}{3} x = -6x
\][/tex]
(because [tex]\(-9 \times \frac{2}{3} = -6\)[/tex])
- The second term:
[tex]\[
(-9) \times 1 = -9
\][/tex]
4. Combine the Terms:
Combining both terms, we get:
[tex]\[
-6x - 9
\][/tex]
Now, let's compare our result, [tex]\(-6x - 9\)[/tex], to the choices 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]
Among these options, the expressions that match [tex]\(-6x - 9\)[/tex] are:
- [tex]\(-9\left(\frac{2}{3} x\right)-9(1)\)[/tex]: This is equivalent after distributing.
- [tex]\(-6x - 9\)[/tex]: This is exactly the simplified form we found.
These two options are equivalent to the original expression [tex]\(-9\left(\frac{2}{3} x+1\right)\)[/tex].