To find the expressions equivalent to \(-9 \left(\frac{2}{3}x + 1\right)\), we will distribute \(-9\) to both terms inside the parentheses:
\[ -9 \left(\frac{2}{3}x + 1\right) = -9 \cdot \frac{2}{3}x - 9 \cdot 1 \]
Calculating the first term: \[ -9 \cdot \frac{2}{3}x = -6x \]
Calculating the second term: \[ -9 \cdot 1 = -9 \]
Putting it all together, we get: \[ -9 \left(\frac{2}{3}x + 1\right) = -6x - 9 \]
Now we’ll compare this result with the given options:
A. \(-9 \left(\frac{2}{3}x\right) + 9 \left(1\right)\)
- This simplifies to \(-6x + 9\), not equivalent.
B. \(-6x + 1\)
- This does not match, not equivalent.
C. \(-9 \left(\frac{2}{3}x\right) - 9 \left(1\right)\)
- This simplifies to \(-6x - 9\), equivalent.
D. \(-6x - 9\)
- This matches our result, equivalent.
E. \(-6x + 9\)
- This does not match, not equivalent.
The two expressions that are equivalent to \(-9 \left(\frac{2}{3}x + 1\right)\) are:
C. -9 (2/3x) - 9 (1) and D. -6x - 9.