MHB Simplify and state any restrictions on the variables.

AI Thread Summary
The discussion focuses on simplifying two algebraic expressions and identifying errors in the simplification process. In the first expression, the correct approach involves factoring out a common term from the numerator rather than expanding it unnecessarily. The second expression also highlights a mistake in the denominator's simplification, where the factorization was incorrect. Participants emphasize the importance of correctly identifying and canceling common factors to achieve the proper simplification. Overall, the thread underscores the need for careful attention to detail in algebraic manipulations.
eleventhxhour
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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:

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

$$\frac{-2}{9x^3} ⋅ 12x^4 $$

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

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

$$\frac{-8x}{9} $$
 
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Re: Simplify

eleventhxhour said:
$$\frac{3a+6}{a+2}$$

$$3 + \frac{6}{2}$$
\[
\frac{x_1+y_1}{x_2+y_2}\ne\frac{x_1}{y_1}+\frac{x_2}{y_2}
\]
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 $a-2$.

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

$$\frac{3x^3(-8x)}{3x^3(9)} $$
The last denominator should be $3x^3\cdot3$ instead of $3x^3\cdot9$.
 
Thanks! That helped a lot.
 
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