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]}

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