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Answer :
We start by expanding the expression:
$$
-9\left(\frac{2}{3}x + 1\right).
$$
Distribute the $-9$ to each term inside the parentheses:
$$
-9 \cdot \frac{2}{3}x - 9 \cdot 1.
$$
Calculate each multiplication:
1. For the first term:
$$
-9 \cdot \frac{2}{3}x = -\frac{18}{3}x = -6x.
$$
2. For the second term:
$$
-9 \cdot 1 = -9.
$$
So, the simplified expression is:
$$
-6x - 9.
$$
Now, compare this with the given options:
1. **Option 1:** $$-9\left(\frac{2}{3} x\right)+9(1) = -6x+9.$$
This is not equivalent to $$-6x-9.$$
2. **Option 2:** $$-9\left(\frac{2}{3} x\right)-9(1) = -6x-9.$$
This matches our simplified expression.
3. **Option 3:** $$-9\left(\frac{2}{3} x\right)+1 = -6x+1.$$
This is not equivalent.
4. **Option 4:** $$-6 x+1.$$
This is not equivalent.
5. **Option 5:** $$-6 x+9.$$
This is not equivalent.
6. **Option 6:** $$-6 x-9.$$
This exactly matches our simplified expression.
Thus, the expressions that are equivalent to
$$
-9\left(\frac{2}{3}x+1\right)
$$
are those in **Option 2** and **Option 6**.
$$
-9\left(\frac{2}{3}x + 1\right).
$$
Distribute the $-9$ to each term inside the parentheses:
$$
-9 \cdot \frac{2}{3}x - 9 \cdot 1.
$$
Calculate each multiplication:
1. For the first term:
$$
-9 \cdot \frac{2}{3}x = -\frac{18}{3}x = -6x.
$$
2. For the second term:
$$
-9 \cdot 1 = -9.
$$
So, the simplified expression is:
$$
-6x - 9.
$$
Now, compare this with the given options:
1. **Option 1:** $$-9\left(\frac{2}{3} x\right)+9(1) = -6x+9.$$
This is not equivalent to $$-6x-9.$$
2. **Option 2:** $$-9\left(\frac{2}{3} x\right)-9(1) = -6x-9.$$
This matches our simplified expression.
3. **Option 3:** $$-9\left(\frac{2}{3} x\right)+1 = -6x+1.$$
This is not equivalent.
4. **Option 4:** $$-6 x+1.$$
This is not equivalent.
5. **Option 5:** $$-6 x+9.$$
This is not equivalent.
6. **Option 6:** $$-6 x-9.$$
This exactly matches our simplified expression.
Thus, the expressions that are equivalent to
$$
-9\left(\frac{2}{3}x+1\right)
$$
are those in **Option 2** and **Option 6**.
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