Answer :
To determine which expressions are equivalent to [tex]\(-9\left(\frac{2}{3} x + 1\right)\)[/tex], we need to simplify the original expression and compare it with the options provided.
### Step 1: Simplify the Original Expression
The expression is [tex]\(-9\left(\frac{2}{3} x + 1\right)\)[/tex]. Let's distribute the [tex]\(-9\)[/tex] to both terms inside the parentheses:
1. First term: [tex]\(-9 \times \frac{2}{3} x = -\frac{18}{3} x = -6x\)[/tex]
2. Second term: [tex]\(-9 \times 1 = -9\)[/tex]
So, the expression simplifies to [tex]\(-6x - 9\)[/tex].
### Step 2: Compare with the Given Options
Let's check each option to see if they match [tex]\(-6x - 9\)[/tex]:
1. [tex]\(-9\left(\frac{2}{3} x\right) + 9(1)\)[/tex]
- This simplifies to: [tex]\(-6x + 9\)[/tex], which is not equivalent to [tex]\(-6x - 9\)[/tex].
2. [tex]\(-9\left(\frac{2}{3} x\right) - 9(1)\)[/tex]
- This simplifies to: [tex]\(-6x - 9\)[/tex], which is exactly equivalent to the simplified expression [tex]\(-6x - 9\)[/tex].
3. [tex]\(-9\left(\frac{2}{3} x\right) + 1\)[/tex]
- This simplifies to: [tex]\(-6x + 1\)[/tex], which is not equivalent to [tex]\(-6x - 9\)[/tex].
4. [tex]\(-6x + 1\)[/tex]
- This is not equivalent to [tex]\(-6x - 9\)[/tex].
5. [tex]\(-6x + 9\)[/tex]
- This is not equivalent to [tex]\(-6x - 9\)[/tex].
6. [tex]\(-6x - 9\)[/tex]
- This matches the simplified version of the original expression, [tex]\(-6x - 9\)[/tex].
### Conclusion
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]
These two options correctly represent the original expression when simplified.
### Step 1: Simplify the Original Expression
The expression is [tex]\(-9\left(\frac{2}{3} x + 1\right)\)[/tex]. Let's distribute the [tex]\(-9\)[/tex] to both terms inside the parentheses:
1. First term: [tex]\(-9 \times \frac{2}{3} x = -\frac{18}{3} x = -6x\)[/tex]
2. Second term: [tex]\(-9 \times 1 = -9\)[/tex]
So, the expression simplifies to [tex]\(-6x - 9\)[/tex].
### Step 2: Compare with the Given Options
Let's check each option to see if they match [tex]\(-6x - 9\)[/tex]:
1. [tex]\(-9\left(\frac{2}{3} x\right) + 9(1)\)[/tex]
- This simplifies to: [tex]\(-6x + 9\)[/tex], which is not equivalent to [tex]\(-6x - 9\)[/tex].
2. [tex]\(-9\left(\frac{2}{3} x\right) - 9(1)\)[/tex]
- This simplifies to: [tex]\(-6x - 9\)[/tex], which is exactly equivalent to the simplified expression [tex]\(-6x - 9\)[/tex].
3. [tex]\(-9\left(\frac{2}{3} x\right) + 1\)[/tex]
- This simplifies to: [tex]\(-6x + 1\)[/tex], which is not equivalent to [tex]\(-6x - 9\)[/tex].
4. [tex]\(-6x + 1\)[/tex]
- This is not equivalent to [tex]\(-6x - 9\)[/tex].
5. [tex]\(-6x + 9\)[/tex]
- This is not equivalent to [tex]\(-6x - 9\)[/tex].
6. [tex]\(-6x - 9\)[/tex]
- This matches the simplified version of the original expression, [tex]\(-6x - 9\)[/tex].
### Conclusion
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]
These two options correctly represent the original expression when simplified.
