Inequalities and System of Inequalities with Two Variables

Consider inequality `f(x,y)>g(x,y)`, which we will call inequality with two variables. Solution of such inequaltity is pair of value of arguments, that convert inequality into correct numerical identity.

It is known that pair of real numbers (x;y) unambiguousy defines point of coordinate line. This gives an ability to draw solutions of inequality or system of inequalities with two variables geometrically, in the form of some set of points of coordinate line.

Example. Draw on coordinate plane set of solutions of the system of inequalities `{(x>=0),(y>=0),(xy>4),(x+y<5):}`.

Geometrically inequalities `x>=0,y>=0` represent set of points that lie in the first quadrant.

Next we draw dashed line `x+y<5` or `y<5-x` (not solid because we have strict inequality). Line divides plane on two regions. We need now to define which region to take. For this take any region and take any point inside it. We take region that lies under line y=5-x and take point (0,0). Plugging these point into equation of line yields `0<5-0`. This is correct inequality, therefore the region we take is under the line.drawing inequalities

Next, since `x>=0` then `xy>4` is equivalent to `y>4/x`. We draw hyperbola (again dashed) and again need to define what region to take (above hyperbola or below). We take region under hyperbola and take point (1,1). Therefore, `1>4/1`. This is incorrect so we need to take region above hyperbola.

Intersection of first quadrant, region below line and region above hyperbola will give required region (set) of solutions (see figure).