Answer :
To determine which expressions are equivalent to [tex]\(-9\left(\frac{2}{3} x+1\right)\)[/tex], we need to simplify the expression by distributing the [tex]\(-9\)[/tex] across the terms inside the parentheses. Here’s how we do it step-by-step:
1. Distribution of [tex]\(-9\)[/tex]:
We have [tex]\(-9\left(\frac{2}{3}x + 1\right)\)[/tex]. We need to distribute [tex]\(-9\)[/tex] to both [tex]\(\frac{2}{3}x\)[/tex] and 1:
- 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 1:
[tex]\[
-9 \times 1 = -9
\][/tex]
2. Combine the results:
Combine the distributed terms:
[tex]\[
-6x - 9
\][/tex]
Now let's compare this with the options given to see which are equivalent:
- [tex]\(-9\left(\frac{2}{3} x\right)+9(1)\)[/tex] simplifies to [tex]\(-6x + 9\)[/tex], which is not equivalent.
- [tex]\(-9\left(\frac{2}{3} x\right)-9(1)\)[/tex] simplifies to [tex]\(-6x - 9\)[/tex], which is equivalent.
- [tex]\(-9\left(\frac{2}{3} x\right)+1\)[/tex] simplifies to [tex]\(-6x + 1\)[/tex], which is not equivalent.
- [tex]\(-6x + 1\)[/tex] is not equivalent.
- [tex]\(-6x + 9\)[/tex] is not equivalent.
- [tex]\(-6x - 9\)[/tex] is equivalent.
Therefore, 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]
1. Distribution of [tex]\(-9\)[/tex]:
We have [tex]\(-9\left(\frac{2}{3}x + 1\right)\)[/tex]. We need to distribute [tex]\(-9\)[/tex] to both [tex]\(\frac{2}{3}x\)[/tex] and 1:
- 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 1:
[tex]\[
-9 \times 1 = -9
\][/tex]
2. Combine the results:
Combine the distributed terms:
[tex]\[
-6x - 9
\][/tex]
Now let's compare this with the options given to see which are equivalent:
- [tex]\(-9\left(\frac{2}{3} x\right)+9(1)\)[/tex] simplifies to [tex]\(-6x + 9\)[/tex], which is not equivalent.
- [tex]\(-9\left(\frac{2}{3} x\right)-9(1)\)[/tex] simplifies to [tex]\(-6x - 9\)[/tex], which is equivalent.
- [tex]\(-9\left(\frac{2}{3} x\right)+1\)[/tex] simplifies to [tex]\(-6x + 1\)[/tex], which is not equivalent.
- [tex]\(-6x + 1\)[/tex] is not equivalent.
- [tex]\(-6x + 9\)[/tex] is not equivalent.
- [tex]\(-6x - 9\)[/tex] is equivalent.
Therefore, 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]