Question 118165

For example, look at the graph of following equality:
{{{2x +3y=6}}}.	

*[invoke Plot_a_graph_of_functions "(-2/3)x+2", -15, 15, -10, 15]

Recall that when a line is drawn in a plane, the line divides the plane into three sets of points:

{{{two}}}{{{ half-planes}}} and the {{{line}}}{{{ itself}}}. 
	
	Now, consider the solution to {{{2x +3y<=6}}}.. 
Recall that the solution is {{{one}}} of these {{{half-planes}}} along with the{{{ line}}}. 

Note that {{{2x +3y<6}}} would result in the same half-plane;{{{ but}}}{{{ not}}}{{{the}}}{{{ line}}}.

One method for {{{determining}}} the {{{correct}}}{{{ half-plane}}} is to "{{{test}}}" the {{{inequality}}} with the coordinates of a point that is {{{NOT}}} on the line. 

If the coordinates {{{satisfy}}} the inequality, then {{{that}}}{{{ point}}}{{{ and}}}{{{ all }}}{{{points}}} in that {{{half-plane}}} are solutions. 

We then {{{shade}}}{{{ that}}}{{{ half-plane}}}. 

Usually the point {{{O}}}=({{{0}}},{{{0}}}) is the simplest {{{test }}}{{{point}}}-provided the line {{{does}}}{{{ not }}}{{{pass}}} through the{{{ origin}}}.

In this example, the ({{{0}}},{{{0}}}) satisfies the inequality and gives us the solution:

On graph above, shade the {{{line}}} and the {{{ half-plane}}} which lies on the left from the line.