Solve the system of inequalities graphically y < x 4x^2-16x-3y^2-12y < 8

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embizze eNotes educator| Certified Educator

We are asked to solve the system of inequalitites graphically:



First graph the inequality `y<x` : this is the line `y=x` , drawn dotted, and shaded "beneath" (the side that includes the point (1,-1)).

Now we consider the other inequality. We know that it is a conic section, perhaps degenerate. Use completing the square to write in standard form:






This is a hyperbola, centered at (2,-2) whose branches open left/right. The area to shade is between the branches.

To draw the hyperbola lightly draw a rectangle whose center is at (2,-2) and whose vertices are `(2+sqrt(3),0),(2-sqrt(3),0),(2+sqrt(3),-4),(2-sqrt(3),-4)`

Then draw the diagonals -- the diagonals are the asymptotes for the branches.

The vertices of the branches are at `(2-sqrt(3),-2),(2+sqrt(3),-2)`

The solution set will include points between the branches that are beneath the line y=x.

** The shaded region is bounded at the top, and unbounded at the bottom.**

(There is an artifact of the grapher -- there should not be lines connecting the branches of the hyperbola)