GRAPHING INEQUALITIES

 Definition 1.
 Consider any of these inequalities: AB , AB , A > B , A < B. For each such inequality, it's ASSOCIATED EQUATION is A = B. We will abbreviate the phrase "ASSOCIATED EQUATION" to "AE".
 Example 2.
 The AE for the inequality   "3x - 2y 5"    is    "3x - 2y = 5"  .
 Fact 3.
Let A and B be algebraic expressions involving the variables x, y, or others.
The graph of the equation A = B divides up space into pieces;
within each piece,
 either [1] ALL points satisfy A > B , or or      [2] ALL points satisfy A < B.
 Note 4.
 In our situations, A and B will usually involve the two variables "x" and "y", the AE's will have straight-line graphs, and therefore the pieces into which A = B divides space will be "half-planes", that is, the set of points on one side of a straight line.
 Note 5.
 To graph an inequality (such as A B), first graph it's AE. Then test each piece into which that graph divides space (with "TEST POINTS": red points in figure below). Choose or mark those pieces whose test points satisfy the given inequality (in our example, A B). Thus, in the examples below, the origin ( 0 , 0 ) makes a good test point, for which the stated inequality " x 2 + y 2 1 " is TRUE.

ARROWS are used when marking several overlapping regions. Thus, the graph at the right marks the interior of the circle with center (0,0) and radius 1, which solves the inequality " x2 + y2 1 ".

 The test points are C = (0,0) and D = (2,0): C satisfies " x2 + y21" since " 02 + 021". On the other hand, D fails to satisfy " x2 + y2 1 ", because " 22 + 02 = 4 > 1 ". Thus, using Fact 3 above, the solution set for our inequality is the set of points on the same side of the circle as is C.