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Equality with absolute value

Fernando Revilla

Well-known member
MHB Math Helper
Jan 29, 2012
661
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

Well-known member
MHB Math Helper
Jan 29, 2012
661
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.