To determine which expressions are equivalent to \(-9 \left(\frac{2}{3}x + 1\right)\), we need to simplify the expression:
\[ -9 \left(\frac{2}{3}x + 1\right) = -9 \cdot \frac{2}{3}x - 9 \cdot 1 \] This simplifies to: \[ -9 \cdot \frac{2}{3}x - 9 = -6x - 9 \]
Now let's evaluate the options given:
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-6x - 9: This is equivalent to our simplified expression, so this is correct.
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0 - 9 \left(\frac{2}{3}x\right) - 9(1): This simplifies to \(-9 \cdot \frac{2}{3}x - 9\), which is equivalent to \(-6x - 9\). So this also applies.
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-6x + 9: This is not equivalent, as it does not match \(-6x - 9\).
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-9 \left(\frac{2}{3}x\right) + 9(1): This simplifies to \(-9 \cdot \frac{2}{3}x + 9\), which is \(-6x + 9\), and this is also not equivalent.
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-6x + 1: This is also not equivalent, as it does not match \(-6x - 9\).
Thus, the two expressions that are equivalent to \(-9 \left(\frac{2}{3}x + 1\right)\) are:
- -6x - 9
- 0 - 9 (2/3x) - 9(1)