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Answer :
To find which expressions are equivalent to [tex]\(-9\left(\frac{2}{3}x + 1\right)\)[/tex], we need to distribute and simplify the expression:
1. Distribute the -9:
We start by distributing the [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]
2. Calculate each term:
- For the first term:
[tex]\[
-9 \cdot \frac{2}{3}x = -6x
\][/tex]
- For the second term:
[tex]\[
-9 \cdot 1 = -9
\][/tex]
3. Combine the results:
So, the expression simplifies to:
[tex]\[
-6x - 9
\][/tex]
Now, let's check which given expressions are equivalent to [tex]\(-6x - 9\)[/tex]:
- -9\left(\frac{2}{3}x\right) + 9(1):
This simplifies to [tex]\(-6x + 9\)[/tex], which is not equivalent.
- -9\left(\frac{2}{3}x\right) - 9(1):
This simplifies to [tex]\(-6x - 9\)[/tex], which is equivalent.
- -9\left(\frac{2}{3}x\right) + 1:
This does not match our simplification because of the +1.
- -6x + 1:
This does not match our simplification.
- -6x + 9:
This does not match our simplification.
- -6x - 9:
This matches our simplified expression exactly.
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. Distribute the -9:
We start by distributing the [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]
2. Calculate each term:
- For the first term:
[tex]\[
-9 \cdot \frac{2}{3}x = -6x
\][/tex]
- For the second term:
[tex]\[
-9 \cdot 1 = -9
\][/tex]
3. Combine the results:
So, the expression simplifies to:
[tex]\[
-6x - 9
\][/tex]
Now, let's check which given expressions are equivalent to [tex]\(-6x - 9\)[/tex]:
- -9\left(\frac{2}{3}x\right) + 9(1):
This simplifies to [tex]\(-6x + 9\)[/tex], which is not equivalent.
- -9\left(\frac{2}{3}x\right) - 9(1):
This simplifies to [tex]\(-6x - 9\)[/tex], which is equivalent.
- -9\left(\frac{2}{3}x\right) + 1:
This does not match our simplification because of the +1.
- -6x + 1:
This does not match our simplification.
- -6x + 9:
This does not match our simplification.
- -6x - 9:
This matches our simplified expression exactly.
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]
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