Help me solve this precalculus algebra expression!

Question

Expression and my attempt at solution: $$\frac{3ab}{c^{-1}}:\left(\frac{b}{c^{-1}}+\frac{a}{c^{-1}}-\frac{a}{b^{-1}}\right)-\frac{(a-1)a^{-1}+(b-1)b^{-1}+(c+1)c^{-1}}{a^{-1}+b^{-1}-c^{-1}}=$$ $$\frac{3ab}{c^{-1}}\cdot\frac{c^{-1}b^{-1}}{a+b-ac^{-1}}-\frac{1-a^{-1}+1-b^{-1}+1+c^{-1}}{a^{-1}+b^{-1}-c^{-1}}=$$ $$\frac{3a}{a+b-ac^{-1}}-\frac{-3(a^{-1}+b^{-1}-c^{-1})}{a^{-1}+b^{-1}-c^{-1}}=$$ $$\frac{3a}{a+b-ac^{-1}}+3=\frac{6a+3b-3ac^{-1}}{a+b-ac^{-1}}=\frac{\frac{6ac}{c}+\frac{3bc}{c}-\frac{3a}{c}}{\frac{ac}{c}+\frac{bc}{c}-\frac{a}{c}}=$$ $$\frac{3(2a+b-a)}{c(a+b)-a}=\frac{3(a+b)}{c(a+b)-a}$$

Correct answer: $1$. I have no idea why my procedure won't give a right solution - it looks accurate. Please help, guys and girls! Thanks in advance!

(Side note: I'm unable to get appropriately large $($ and $)$, although LaTeX I used is correct.)

Answer 1

At first we have $$\frac{3ab}{c^{-1}}=3abc$$ and then $$\frac{b}{c^{-1}}+\frac{a}{c^{-1}}-\frac{a}{b^{-1}}=bc+ac-ab$$ so we get $$\frac{3abc}{bc+ac-ab}$$ then we have $$\frac{\frac{a-1}{a}+\frac{b-1}{b}+\frac{c+1}{c}}{\frac{bc+ac-ab}{abc}}$$ simplifying this term we get $$\frac{3abc-ac-bc+ab}{bc+ac-ab}$$ Putting things together we obtain $$\frac{ac+bc-ab}{bc+ac-ab}=1$$

Answer 2

Another answer has explained the correct solution.

Here's why you didn't get the correct solution.

You wrote

$$\frac{3ab}{c^{-1}}:\left(\frac{b}{c^{-1}}+\frac{a}{c^{-1}}-\frac{a}{b^{-1}}\right)-\frac{(a-1)a^{-1}+(b-1)b^{-1}+(c+1)c^{-1}}{a^{-1}+b^{-1}-c^{-1}}=$$ $$\frac{3ab}{c^{-1}}\cdot\frac{c^{-1}b^{-1}}{a+b-ac^{-1}}-\frac{1-a^{-1}+1-b^{-1}+1+c^{-1}}{a^{-1}+b^{-1}-c^{-1}}=$$

$$\frac{3a}{a+b-ac^{-1}}+3$$

but you should have written

$$\frac{3ab}{c^{-1}}\cdot\frac{c^{-1}b^{-1}}{\color{red}{1+ab^{-1}}-ac^{-1}}-\frac{1-a^{-1}+1-b^{-1}+1+c^{-1}}{a^{-1}+b^{-1}-c^{-1}}=$$

$$\frac{3a}{\color{red}{1+ab^{-1}}-ac^{-1}} -\frac{\color{orange}3-(a^{-1}+b^{-1}-c^{-1})}{a^{-1}+b^{-1}-c^{-1}}$$

$$=\frac3{\color{red}{a^{-1}+b^{-1}}-c^{-1}}-\frac{\color{orange}3}{a^{-1}+b^{-1}-c^{-1}}+1$$

$$=1.$$