How do you determine whether the pair (-6,-9) is a solution to #y<= (x^2-6)/x#?

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Answer 1 · 2017-01-11T08:04:45

The region of points (x, y), with #y<=x-6/x# is shaded in the graph. #(-6,-9 #) in #Q_3# is well inside.

The given bordering curve #x(y-x)=-6# is a hyperbola, contained

between its asymptotes x = 0 and x = y.

Algebraically, when x = -6,

y on the bordering hyperbola #y=x-6/x=-5 > 9#.

So, the point #(-6, -9)# is below #(-6, -5)#.

Illustrative graph is inserted.

graph{y-x+6/x<=0 [-30, 30, -15, 15]}