Answer :
To solve the problem of finding expressions equivalent to [tex]\(-9\left(\frac{2}{3} x + 1\right)\)[/tex], let's break it down step by step.
We are given the expression [tex]\(-9\left(\frac{2}{3} x + 1\right)\)[/tex] and need to distribute the [tex]\(-9\)[/tex] to both terms inside the parentheses:
1. Distribute [tex]\(-9\)[/tex] to the first term [tex]\(\frac{2}{3} x\)[/tex]:
[tex]\[
-9 \times \frac{2}{3} x = -6x
\][/tex]
2. Distribute [tex]\(-9\)[/tex] to the second term [tex]\(1\)[/tex]:
[tex]\[
-9 \times 1 = -9
\][/tex]
So, the expression [tex]\(-9\left(\frac{2}{3} x + 1\right)\)[/tex] simplifies to:
[tex]\[
-6x - 9
\][/tex]
Now, let's check the options and see which ones match this simplified expression:
- Option 1: [tex]\(-9\left(\frac{2}{3} x\right) + 9(1)\)[/tex] simplifies to [tex]\(-6x + 9\)[/tex]. This is not equivalent to [tex]\(-6x - 9\)[/tex].
- Option 2: [tex]\(-9\left(\frac{2}{3} x\right) - 9(1)\)[/tex] simplifies to [tex]\(-6x - 9\)[/tex]. This matches our simplified expression.
- Option 3: [tex]\(-9\left(\frac{2}{3} x\right) + 1\)[/tex] simplifies to [tex]\(-6x + 1\)[/tex]. This is not equivalent to [tex]\(-6x - 9\)[/tex].
- Option 4: [tex]\(-6x + 1\)[/tex], which is not equivalent to [tex]\(-6x - 9\)[/tex].
- Option 5: [tex]\(-6x + 9\)[/tex], which is not equivalent to [tex]\(-6x - 9\)[/tex].
- Option 6: [tex]\(-6x - 9\)[/tex], which matches our simplified expression exactly.
Therefore, the expressions 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]
We are given the expression [tex]\(-9\left(\frac{2}{3} x + 1\right)\)[/tex] and need to distribute the [tex]\(-9\)[/tex] to both terms inside the parentheses:
1. Distribute [tex]\(-9\)[/tex] to the first term [tex]\(\frac{2}{3} x\)[/tex]:
[tex]\[
-9 \times \frac{2}{3} x = -6x
\][/tex]
2. Distribute [tex]\(-9\)[/tex] to the second term [tex]\(1\)[/tex]:
[tex]\[
-9 \times 1 = -9
\][/tex]
So, the expression [tex]\(-9\left(\frac{2}{3} x + 1\right)\)[/tex] simplifies to:
[tex]\[
-6x - 9
\][/tex]
Now, let's check the options and see which ones match this simplified expression:
- Option 1: [tex]\(-9\left(\frac{2}{3} x\right) + 9(1)\)[/tex] simplifies to [tex]\(-6x + 9\)[/tex]. This is not equivalent to [tex]\(-6x - 9\)[/tex].
- Option 2: [tex]\(-9\left(\frac{2}{3} x\right) - 9(1)\)[/tex] simplifies to [tex]\(-6x - 9\)[/tex]. This matches our simplified expression.
- Option 3: [tex]\(-9\left(\frac{2}{3} x\right) + 1\)[/tex] simplifies to [tex]\(-6x + 1\)[/tex]. This is not equivalent to [tex]\(-6x - 9\)[/tex].
- Option 4: [tex]\(-6x + 1\)[/tex], which is not equivalent to [tex]\(-6x - 9\)[/tex].
- Option 5: [tex]\(-6x + 9\)[/tex], which is not equivalent to [tex]\(-6x - 9\)[/tex].
- Option 6: [tex]\(-6x - 9\)[/tex], which matches our simplified expression exactly.
Therefore, the expressions 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]
