Which expressions are equivalent to −9(2 over 3x+1)

Question

? Check the two that apply.(2 points)
Responses

−6x+9
negative 6 x plus 9

−6x+1
negative 6 x plus 1

−6x−9
negative 6 x minus 9

−9(23x)−9(1)
negative 9 times open paren 2 thirds x close paren minus 9 times 1

−9(23x)+9(1)
negative 9 times open paren 2 thirds x close paren plus 9 times 1

Answer 1

To find expressions equivalent to \( -9\left(\frac{2}{3}x + 1\right) \), we can distribute \( -9 \) through the expression:

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

Calculating \( -9\left(\frac{2}{3}x\right) \):

\[ -9\left(\frac{2}{3}x\right) = -\frac{18}{3}x = -6x \]

And calculating \( -9(1) \):

\[ -9(1) = -9 \]

Putting it all together, we have:

\[ -9\left(\frac{2}{3}x + 1\right) = -6x - 9 \]

Now, checking the response options:

1. \( -6x + 9 \) - Not Equivalent
2. \( -6x + 1 \) - Not Equivalent
3. \( -6x - 9 \) - Equivalent
4. \( -9(23x) - 9(1) \) - Not Equivalent, since it contains a mistake in the notation of \( \frac{2}{3} \).
5. \( -9(23x) + 9(1) \) - Not Equivalent, for the same reason as above.

Thus, the only equivalent expression is:

• \( -6x - 9 \)

The one that applies is:

• \( -6x - 9 \)

Answer 2

are you sure bout that?

Answer 3

Let’s carefully review the problem to ensure accuracy in identifying equivalent expressions to \( -9\left(\frac{2}{3}x + 1\right) \).

Starting with the expression:

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

We can distribute \( -9 \):

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

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

\[ -9 \cdot \frac{2}{3} = -\frac{18}{3} = -6 \]

Thus:

\[ -9\left(\frac{2}{3}x + 1\right) = -6x - 9 \]

Now, let’s examine the choices provided:

1. \( -6x + 9 \) - Not equivalent
2. \( -6x + 1 \) - Not equivalent
3. \( -6x - 9 \) - Equivalent
4. \( -9(2/3x) - 9(1) \) - Correctly written as \( -9 \cdot \frac{2}{3}x - 9 \), which simplifies to \( -6x - 9 \) - Equivalent
5. \( -9(2/3x) + 9(1) \) - Would simplify to \( -6x + 9 \) - Not equivalent

So the two equivalent expressions are:

• \( -6x - 9 \)
• The one resembling \( -9\left(\frac{2}{3}x\right) - 9 \) (if correctly noted as \( -9\left(\frac{2}{3}x\right) - 9 \) rather than \( -9(2/3x) \pm 9(1) \)).

If you were referring to the exact notation in choice 4, it needs to be corrected to present equivalently as stated.

In summary, the confirmed equivalent expressions are:

• \( -6x - 9 \)
• The distribution breakdown of \( -9\left(\frac{2}{3}x\right) - 9(1) = -6x - 9 \) if specified clearly.

Let me know if you need any further clarification!