Solving Absolute Value Inequalities: Steps & Link to Yahoo! Answers

[Fernando Revilla]

I quote a question from Yahoo! Answers

Solve the following inequaltities: a - l 1/bxy l = b
l = absolute value
Please include all the steps. Thank you!

I have given a link to the topic there so the OP can see my response.

 

[Fernando Revilla]

I suppose you mean: solve the equality $a - \left| \dfrac{1}{bxy}\right| = b$. In such case, necessarily $b\ne 0$ and $xy\ne 0$. Denote $D_1=\{(x,y)\in\mathbb{R}^2:x>0,y>0\}$ the open first quadrant, then $$a - \left| \dfrac{1}{bxy}\right| = b\Leftrightarrow a-\frac{1}{|b|xy}=b\Leftrightarrow y=\frac{1}{(a-b)|b|}\frac{1}{x}$$
If $a>b$ we get a branch of an equilateral hyperbola on $D_1$. If $a\le b$, the empty set. You can follow similar arguments for the rest of open quadrants.