Answer :
Let's simplify the expression [tex]\(-9\left(\frac{2}{3} x + 1\right)\)[/tex] step-by-step to identify which of the given expressions are equivalent to it.
1. Distribute the [tex]\(-9\)[/tex] into the expression inside the parentheses:
[tex]\[
-9\left(\frac{2}{3} x + 1\right) = -9 \cdot \frac{2}{3} x + (-9) \cdot 1
\][/tex]
2. Calculate each part of the distribution:
- [tex]\( -9 \cdot \frac{2}{3} x = -\frac{18}{3} x = -6x \)[/tex]
- [tex]\((-9) \cdot 1 = -9\)[/tex]
3. Combine the results:
[tex]\[
-6x - 9
\][/tex]
Now, let's compare this simplified expression [tex]\(-6x - 9\)[/tex] with the given choices:
- Expression 1: [tex]\(-9\left(\frac{2}{3} x\right) + 9(1)\)[/tex]
- Simplifies to: [tex]\(-6x + 9\)[/tex]
- Not equivalent
- Expression 2: [tex]\(-9\left(\frac{2}{3} x\right) - 9(1)\)[/tex]
- Simplifies to: [tex]\(-6x - 9\)[/tex]
- Equivalent
- Expression 3: [tex]\(-9\left(\frac{2}{3} x\right) + 1\)[/tex]
- Simplifies to: [tex]\(-6x + 1\)[/tex]
- Not equivalent
- Expression 4: [tex]\(-6x + 1\)[/tex]
- Not equivalent
- Expression 5: [tex]\(-6x + 9\)[/tex]
- Not equivalent
- Expression 6: [tex]\(-6x - 9\)[/tex]
- Equivalent
Thus, 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. Distribute the [tex]\(-9\)[/tex] into the expression inside the parentheses:
[tex]\[
-9\left(\frac{2}{3} x + 1\right) = -9 \cdot \frac{2}{3} x + (-9) \cdot 1
\][/tex]
2. Calculate each part of the distribution:
- [tex]\( -9 \cdot \frac{2}{3} x = -\frac{18}{3} x = -6x \)[/tex]
- [tex]\((-9) \cdot 1 = -9\)[/tex]
3. Combine the results:
[tex]\[
-6x - 9
\][/tex]
Now, let's compare this simplified expression [tex]\(-6x - 9\)[/tex] with the given choices:
- Expression 1: [tex]\(-9\left(\frac{2}{3} x\right) + 9(1)\)[/tex]
- Simplifies to: [tex]\(-6x + 9\)[/tex]
- Not equivalent
- Expression 2: [tex]\(-9\left(\frac{2}{3} x\right) - 9(1)\)[/tex]
- Simplifies to: [tex]\(-6x - 9\)[/tex]
- Equivalent
- Expression 3: [tex]\(-9\left(\frac{2}{3} x\right) + 1\)[/tex]
- Simplifies to: [tex]\(-6x + 1\)[/tex]
- Not equivalent
- Expression 4: [tex]\(-6x + 1\)[/tex]
- Not equivalent
- Expression 5: [tex]\(-6x + 9\)[/tex]
- Not equivalent
- Expression 6: [tex]\(-6x - 9\)[/tex]
- Equivalent
Thus, 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]