Question

Which expressions are equivalent to -9 (2/3x + 1) Check the two that apply.
(2 points)
• -6x - 9
-9 (2/3x) - 9(1)
* -6x + 9
* -9 (2/3x) + 9 (1)
* -6x + 1

Answer 1

To determine which expressions are equivalent to \(-9 \left( \frac{2}{3}x + 1 \right)\), we'll first simplify this expression.

1. Distribute \(-9\):

\[
-9 \left( \frac{2}{3}x + 1 \right) = -9 \cdot \frac{2}{3}x - 9 \cdot 1
\]

2. Calculate \(-9 \cdot \frac{2}{3}x\):

\[
-9 \cdot \frac{2}{3} = -6
\]
So this part becomes \(-6x\).

3. 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 \(-9 \left( \frac{2}{3}x + 1 \right)\):

1. **-6x - 9** (This is equivalent, as shown.)
2. **-9 (2/3x) - 9(1)** (This is equivalent as it expands to \(-6x - 9\).)
3. **-6x + 9** (Not equivalent, since this would equal \(-6x\) plus a positive \(9\).)
4. **-9 (2/3x) + 9 (1)** (This simplifies to \(-6x + 9\), not equivalent.)
5. **-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)**