To determine which expressions are equivalent to \(-9 \left( \frac{2}{3}x + 1 \right)\), we'll first simplify this expression.
- Distribute \(-9\):
\[ -9 \left( \frac{2}{3}x + 1 \right) = -9 \cdot \frac{2}{3}x - 9 \cdot 1 \]
- Calculate \(-9 \cdot \frac{2}{3}x\):
\[ -9 \cdot \frac{2}{3} = -6 \] So this part becomes \(-6x\).
- Now subtract \(9\):
\[ -6x - 9 \]
Thus, the original expression simplifies to:
\[ -6x - 9 \]
Now let's compare this with each of the given options to find which ones are equivalent to \(-9 \left( \frac{2}{3}x + 1 \right)\):
- -6x - 9 (This is equivalent, as shown.)
- -9 (2/3x) - 9(1) (This is equivalent as it expands to \(-6x - 9\).)
- -6x + 9 (Not equivalent, since this would equal \(-6x\) plus a positive \(9\).)
- -9 (2/3x) + 9 (1) (This simplifies to \(-6x + 9\), not equivalent.)
- -6x + 1 (Not equivalent, as it does not match.)
So, the two expressions that are equivalent to \(-9 \left( \frac{2}{3}x + 1 \right)\) are:
- \(-6x - 9\)
- \(-9 (2/3x) - 9(1)\)
Therefore the correct selections are:
- -6x - 9
- -9 (2/3x) - 9(1)