Answer :
Sure, I can help explain why certain expressions are equivalent to the original expression [tex]\(-9\left(\frac{2}{3} \pi+1\right)\)[/tex].
Let's start by simplifying the original expression step-by-step:
1. Distribute the Multiplication:
[tex]\[
-9\left(\frac{2}{3} \pi + 1\right) = -9 \cdot \frac{2}{3} \pi - 9 \cdot 1
\][/tex]
2. Calculate Each Part:
- The first part is: [tex]\(-9 \cdot \frac{2}{3} \pi = -6\pi\)[/tex]
- The second part is: [tex]\(-9 \cdot 1 = -9\)[/tex]
3. Combine the Parts:
[tex]\[
-6\pi - 9
\][/tex]
Now let's analyze each given expression to see if it matches [tex]\(-6x - 9\)[/tex] when [tex]\(x = \pi\)[/tex]:
- [tex]\(-9\left(\frac{2}{3} x\right)+9(1)\)[/tex]: This evaluates to [tex]\(-6x + 9\)[/tex], which is not equivalent to [tex]\(-6x - 9\)[/tex].
- [tex]\(-9\left(\frac{2}{3} x\right)-9(1)\)[/tex]: This evaluates to [tex]\(-6x - 9\)[/tex], which matches our simplified expression.
- [tex]\(-9\left(\frac{2}{3} x\right)+1\)[/tex]: This evaluates to [tex]\(-6x + 1\)[/tex], which does not match.
- [tex]\(-6x+1\)[/tex]: This is clearly not equal to [tex]\(-6x - 9\)[/tex].
- [tex]\(-6x+9\)[/tex]: This also does not match [tex]\(-6x - 9\)[/tex].
- [tex]\(-6x-9\)[/tex]: This expression matches exactly with our simplified expression.
Based on this breakdown, the expressions that are equivalent to [tex]\(-9\left(\frac{2}{3} \pi+1\right)\)[/tex] are:
- [tex]\(-9\left(\frac{2}{3} x\right)-9(1)\)[/tex]
- [tex]\(-6x-9\)[/tex]
These are the expressions that simplify to [tex]\(-6x - 9\)[/tex] when substituting [tex]\(x = \pi\)[/tex].
Let's start by simplifying the original expression step-by-step:
1. Distribute the Multiplication:
[tex]\[
-9\left(\frac{2}{3} \pi + 1\right) = -9 \cdot \frac{2}{3} \pi - 9 \cdot 1
\][/tex]
2. Calculate Each Part:
- The first part is: [tex]\(-9 \cdot \frac{2}{3} \pi = -6\pi\)[/tex]
- The second part is: [tex]\(-9 \cdot 1 = -9\)[/tex]
3. Combine the Parts:
[tex]\[
-6\pi - 9
\][/tex]
Now let's analyze each given expression to see if it matches [tex]\(-6x - 9\)[/tex] when [tex]\(x = \pi\)[/tex]:
- [tex]\(-9\left(\frac{2}{3} x\right)+9(1)\)[/tex]: This evaluates to [tex]\(-6x + 9\)[/tex], which is not equivalent to [tex]\(-6x - 9\)[/tex].
- [tex]\(-9\left(\frac{2}{3} x\right)-9(1)\)[/tex]: This evaluates to [tex]\(-6x - 9\)[/tex], which matches our simplified expression.
- [tex]\(-9\left(\frac{2}{3} x\right)+1\)[/tex]: This evaluates to [tex]\(-6x + 1\)[/tex], which does not match.
- [tex]\(-6x+1\)[/tex]: This is clearly not equal to [tex]\(-6x - 9\)[/tex].
- [tex]\(-6x+9\)[/tex]: This also does not match [tex]\(-6x - 9\)[/tex].
- [tex]\(-6x-9\)[/tex]: This expression matches exactly with our simplified expression.
Based on this breakdown, the expressions that are equivalent to [tex]\(-9\left(\frac{2}{3} \pi+1\right)\)[/tex] are:
- [tex]\(-9\left(\frac{2}{3} x\right)-9(1)\)[/tex]
- [tex]\(-6x-9\)[/tex]
These are the expressions that simplify to [tex]\(-6x - 9\)[/tex] when substituting [tex]\(x = \pi\)[/tex].