Question

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

Answer 1

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

Explanation:

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]}