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

Jan 11, 2017

The region of points (x, y), with $y \le x - \frac{6}{x}$ is shaded in the graph. (-6,-9 ) in ${Q}_{3}$ is well inside.

Explanation:

The given bordering curve $x \left(y - x\right) = - 6$ is a hyperbola, contained

between its asymptotes x = 0 and x = y.

Algebraically, when x = -6,

y on the bordering hyperbola $y = x - \frac{6}{x} = - 5 > 9$.

So, the point $\left(- 6 , - 9\right)$ is below $\left(- 6 , - 5\right)$.

Illustrative graph is inserted.

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