Answer :
To determine which expressions are equivalent to [tex]\(-9\left(\frac{2}{3} x+1\right)\)[/tex], let's transform this expression step-by-step:
1. Distribute [tex]\(-9\)[/tex] into the expression:
[tex]\[
-9\left(\frac{2}{3} x + 1\right) = -9 \cdot \frac{2}{3} x - 9 \cdot 1
\][/tex]
2. Calculate each term separately:
- Multiply [tex]\(-9\)[/tex] by [tex]\(\frac{2}{3} x\)[/tex]:
[tex]\[
-9 \cdot \frac{2}{3} x = -6x
\][/tex]
- Multiply [tex]\(-9\)[/tex] by [tex]\(1\)[/tex]:
[tex]\[
-9 \cdot 1 = -9
\][/tex]
3. Combine the results:
The expanded expression is:
[tex]\[
-6x - 9
\][/tex]
With the expanded expression [tex]\(-6x - 9\)[/tex], let's check each of the given options to see if they are equivalent:
- Option 1: [tex]\(-9\left(\frac{2}{3} x\right) + 9(1)\)[/tex]
Simplifies to:
[tex]\(-6x + 9\)[/tex]
Not equivalent since [tex]\(-6x + 9 \neq -6x - 9\)[/tex].
- Option 2: [tex]\(-9\left(\frac{2}{3} x\right) - 9(1)\)[/tex]
Simplifies to:
[tex]\(-6x - 9\)[/tex]
This is equivalent to [tex]\(-6x - 9\)[/tex].
- Option 3: [tex]\(-9\left(\frac{2}{3} x\right) + 1\)[/tex]
Simplifies to:
[tex]\(-6x + 1\)[/tex]
Not equivalent since [tex]\(-6x + 1 \neq -6x - 9\)[/tex].
- Option 4: [tex]\(-6x + 1\)[/tex]
Not equivalent since [tex]\(-6x + 1 \neq -6x - 9\)[/tex].
- Option 5: [tex]\(-6x + 9\)[/tex]
Not equivalent since [tex]\(-6x + 9 \neq -6x - 9\)[/tex].
- Option 6: [tex]\(-6x - 9\)[/tex]
This is exactly the same expression as our expanded form.
Therefore, the expressions equivalent to [tex]\(-9\left(\frac{2}{3} x+1\right)\)[/tex] are option 2 and option 6:
- [tex]\(-9\left(\frac{2}{3} x\right) - 9(1)\)[/tex]
- [tex]\(-6 x - 9\)[/tex]
1. Distribute [tex]\(-9\)[/tex] into the expression:
[tex]\[
-9\left(\frac{2}{3} x + 1\right) = -9 \cdot \frac{2}{3} x - 9 \cdot 1
\][/tex]
2. Calculate each term separately:
- Multiply [tex]\(-9\)[/tex] by [tex]\(\frac{2}{3} x\)[/tex]:
[tex]\[
-9 \cdot \frac{2}{3} x = -6x
\][/tex]
- Multiply [tex]\(-9\)[/tex] by [tex]\(1\)[/tex]:
[tex]\[
-9 \cdot 1 = -9
\][/tex]
3. Combine the results:
The expanded expression is:
[tex]\[
-6x - 9
\][/tex]
With the expanded expression [tex]\(-6x - 9\)[/tex], let's check each of the given options to see if they are equivalent:
- Option 1: [tex]\(-9\left(\frac{2}{3} x\right) + 9(1)\)[/tex]
Simplifies to:
[tex]\(-6x + 9\)[/tex]
Not equivalent since [tex]\(-6x + 9 \neq -6x - 9\)[/tex].
- Option 2: [tex]\(-9\left(\frac{2}{3} x\right) - 9(1)\)[/tex]
Simplifies to:
[tex]\(-6x - 9\)[/tex]
This is equivalent to [tex]\(-6x - 9\)[/tex].
- Option 3: [tex]\(-9\left(\frac{2}{3} x\right) + 1\)[/tex]
Simplifies to:
[tex]\(-6x + 1\)[/tex]
Not equivalent since [tex]\(-6x + 1 \neq -6x - 9\)[/tex].
- Option 4: [tex]\(-6x + 1\)[/tex]
Not equivalent since [tex]\(-6x + 1 \neq -6x - 9\)[/tex].
- Option 5: [tex]\(-6x + 9\)[/tex]
Not equivalent since [tex]\(-6x + 9 \neq -6x - 9\)[/tex].
- Option 6: [tex]\(-6x - 9\)[/tex]
This is exactly the same expression as our expanded form.
Therefore, the expressions equivalent to [tex]\(-9\left(\frac{2}{3} x+1\right)\)[/tex] are option 2 and option 6:
- [tex]\(-9\left(\frac{2}{3} x\right) - 9(1)\)[/tex]
- [tex]\(-6 x - 9\)[/tex]
