Answer :
To determine which expressions are equivalent to [tex]\(-9\left(\frac{2}{3} x+1\right)\)[/tex], let's simplify and check each option step-by-step.
First, expand and simplify the expression [tex]\(-9\left(\frac{2}{3} x + 1\right)\)[/tex]:
1. Distribute [tex]\(-9\)[/tex] to both terms inside the parentheses:
[tex]\[
-9 \times \frac{2}{3} x + (-9 \times 1)
\][/tex]
2. Calculate each part:
- [tex]\(-9 \times \frac{2}{3} x = -6x\)[/tex]
- [tex]\(-9 \times 1 = -9\)[/tex]
3. Combine these results to get the simplified expression:
[tex]\[
-6x - 9
\][/tex]
Now, compare this simplified form [tex]\(-6x - 9\)[/tex] with the expressions provided:
1. [tex]\(-9\left(\frac{2}{3} x\right) + 9(1)\)[/tex]
- Simplifies to [tex]\(-6x + 9\)[/tex], which does not match [tex]\(-6x - 9\)[/tex].
2. [tex]\(-9\left(\frac{2}{3} x\right) - 9(1)\)[/tex]
- Simplifies to [tex]\(-6x - 9\)[/tex], which matches [tex]\(-6x - 9\)[/tex].
3. [tex]\(-9\left(\frac{2}{3} x\right) + 1\)[/tex]
- Simplifies to [tex]\(-6x + 1\)[/tex], which does not match [tex]\(-6x - 9\)[/tex].
4. [tex]\(-6x + 1\)[/tex]
- Does not match [tex]\(-6x - 9\)[/tex].
5. [tex]\(-6x + 9\)[/tex]
- Does not match [tex]\(-6x - 9\)[/tex].
6. [tex]\(-6x - 9\)[/tex]
- Directly matches [tex]\(-6x - 9\)[/tex].
Therefore, the expressions that are equivalent to the original expression [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]
These two expressions are equivalent to the original expanded and simplified form.
First, expand and simplify the expression [tex]\(-9\left(\frac{2}{3} x + 1\right)\)[/tex]:
1. Distribute [tex]\(-9\)[/tex] to both terms inside the parentheses:
[tex]\[
-9 \times \frac{2}{3} x + (-9 \times 1)
\][/tex]
2. Calculate each part:
- [tex]\(-9 \times \frac{2}{3} x = -6x\)[/tex]
- [tex]\(-9 \times 1 = -9\)[/tex]
3. Combine these results to get the simplified expression:
[tex]\[
-6x - 9
\][/tex]
Now, compare this simplified form [tex]\(-6x - 9\)[/tex] with the expressions provided:
1. [tex]\(-9\left(\frac{2}{3} x\right) + 9(1)\)[/tex]
- Simplifies to [tex]\(-6x + 9\)[/tex], which does not match [tex]\(-6x - 9\)[/tex].
2. [tex]\(-9\left(\frac{2}{3} x\right) - 9(1)\)[/tex]
- Simplifies to [tex]\(-6x - 9\)[/tex], which matches [tex]\(-6x - 9\)[/tex].
3. [tex]\(-9\left(\frac{2}{3} x\right) + 1\)[/tex]
- Simplifies to [tex]\(-6x + 1\)[/tex], which does not match [tex]\(-6x - 9\)[/tex].
4. [tex]\(-6x + 1\)[/tex]
- Does not match [tex]\(-6x - 9\)[/tex].
5. [tex]\(-6x + 9\)[/tex]
- Does not match [tex]\(-6x - 9\)[/tex].
6. [tex]\(-6x - 9\)[/tex]
- Directly matches [tex]\(-6x - 9\)[/tex].
Therefore, the expressions that are equivalent to the original expression [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]
These two expressions are equivalent to the original expanded and simplified form.
