Answer :
To determine which expressions are equivalent to [tex]\(-9\left(\frac{2}{3} x + 1\right)\)[/tex], we need to simplify and distribute the expression step-by-step.
1. Start with the original expression:
[tex]\[
-9\left(\frac{2}{3} x + 1\right)
\][/tex]
2. Distribute the [tex]\(-9\)[/tex] inside the parentheses:
- 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 [tex]\(1\)[/tex]:
[tex]\[
-9 \times 1 = -9
\][/tex]
3. Combine the results:
[tex]\[
-6x - 9
\][/tex]
Now, let's check which of the given options are equivalent to [tex]\(-6x - 9\)[/tex]:
- Option 1: [tex]\(-9\left(\frac{2}{3} x\right) + 9(1)\)[/tex]
- Simplifies to: [tex]\(-6x + 9\)[/tex]. Not equivalent because the constant term is [tex]\(+9\)[/tex] instead of [tex]\(-9\)[/tex].
- Option 2: [tex]\(-9\left(\frac{2}{3} x\right) - 9(1)\)[/tex]
- Simplifies to: [tex]\(-6x - 9\)[/tex]. Equivalent.
- Option 3: [tex]\(-9\left(\frac{2}{3} x\right) + 1\)[/tex]
- Simplifies to: [tex]\(-6x + 1\)[/tex]. Not equivalent because the constant term is [tex]\(+1\)[/tex] instead of [tex]\(-9\)[/tex].
- Option 4: [tex]\(-6x + 1\)[/tex]
- This is not equivalent because the constant term is [tex]\(+1\)[/tex] instead of [tex]\(-9\)[/tex].
- Option 5: [tex]\(-6x + 9\)[/tex]
- This is not equivalent because the constant term is [tex]\(+9\)[/tex] instead of [tex]\(-9\)[/tex].
- Option 6: [tex]\(-6x - 9\)[/tex]
- This is exactly our simplified expression. 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. Start with the original expression:
[tex]\[
-9\left(\frac{2}{3} x + 1\right)
\][/tex]
2. Distribute the [tex]\(-9\)[/tex] inside the parentheses:
- 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 [tex]\(1\)[/tex]:
[tex]\[
-9 \times 1 = -9
\][/tex]
3. Combine the results:
[tex]\[
-6x - 9
\][/tex]
Now, let's check which of the given options are equivalent to [tex]\(-6x - 9\)[/tex]:
- Option 1: [tex]\(-9\left(\frac{2}{3} x\right) + 9(1)\)[/tex]
- Simplifies to: [tex]\(-6x + 9\)[/tex]. Not equivalent because the constant term is [tex]\(+9\)[/tex] instead of [tex]\(-9\)[/tex].
- Option 2: [tex]\(-9\left(\frac{2}{3} x\right) - 9(1)\)[/tex]
- Simplifies to: [tex]\(-6x - 9\)[/tex]. Equivalent.
- Option 3: [tex]\(-9\left(\frac{2}{3} x\right) + 1\)[/tex]
- Simplifies to: [tex]\(-6x + 1\)[/tex]. Not equivalent because the constant term is [tex]\(+1\)[/tex] instead of [tex]\(-9\)[/tex].
- Option 4: [tex]\(-6x + 1\)[/tex]
- This is not equivalent because the constant term is [tex]\(+1\)[/tex] instead of [tex]\(-9\)[/tex].
- Option 5: [tex]\(-6x + 9\)[/tex]
- This is not equivalent because the constant term is [tex]\(+9\)[/tex] instead of [tex]\(-9\)[/tex].
- Option 6: [tex]\(-6x - 9\)[/tex]
- This is exactly our simplified expression. 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]
