Simplify and state any restrictions on the variables.

[eleventhxhour]

Simplify

1) Simplify and state any restrictions on the variables:

$$\frac{3a-6}{a+2} ÷ \frac{a-2}{a+2} $$

This is what I did. Can someone tell me what I did wrong? Thanks.

$$\frac{3a-6}{a+2} ⋅ \frac{a+2}{a-2} $$

$$\frac{3a^2-12}{a^2-4}$$

$$\frac{(3a+6)(a-2)}{(a+2)(a-2)}$$

$$\frac{3a+6}{a+2}$$

$$3 + \frac{6}{2}$$

$$= 6$$

Here is another one where I got the wrong answer, but I'm not sure what I did wrong:

2) Simplify and state any restrictions on the variables:
$$\frac{2(x-2)}{9x^3} ⋅ \frac{12x^4}{2-x} $$

This is what I did:

$$\frac{-2(-x+2)}{9x^3} ⋅ \frac{12x^4}{2-x} $$

$$\frac{-2}{9x^3} ⋅ 12x^4 $$

$$\frac{-24x^4}{9x^3} $$

$$\frac{3x^3(-8x)}{3x^3(9)} $$

$$\frac{-8x}{9} $$

 

[Evgeny.Makarov]

Re: Simplify

$$\frac{3a+6}{a+2}$$

$$3 + \frac{6}{2}$$

\[
\frac{x_1+y_1}{x_2+y_2}\ne\frac{x_1}{y_1}+\frac{x_2}{y_2}
\]
Also, there is no reason to convert $(3a-6)(a+2)$ to $3a^2-12$ and then factor it again as $(3a+6)(a-2)$. Instead, you should have factored 3 out of $3a-6$ and cancel the resulting $a-2$.

$$\frac{-24x^4}{9x^3} $$

$$\frac{3x^3(-8x)}{3x^3(9)} $$

The last denominator should be $3x^3\cdot3$ instead of $3x^3\cdot9$.

 

[eleventhxhour]

Thanks! That helped a lot.