o Boundary Lines
-> Replace each inequality sign with = to get a boundary.
-> Solid boundary: <= or >= includes the line.
-> Dashed boundary: < or > excludes the line.
-> Shade the half-plane that satisfies the inequality.
-> Test a point not on the boundary.
-> Example: y > 2x - 3.
-> Test (0, 0): 0 > -3 is true, so shade above.
-> Example: x < 2 uses a dashed vertical boundary at x = 2.
-> Points on x = 2 are excluded; shade left.
o Sign Reversal
-> Multiplying or dividing by a negative reverses the sign.
-> Example: -3y <= 12 becomes y >= -4.
-> Dividing by -3 flips <= to >=.
-> Example: 2y > -4x + 8.
-> Divide by 2: y > -2x + 4, with the sign unchanged.
-> Rewrite 5 - 2y < x by isolating y carefully.
-> -2y < x - 5, then y > -0.5x + 2.5.
-> The division by -2 flips < to >.
-> Test (0, 3): 5 - 6 = -1 < 0 is true.
o Overlap
-> The feasible region satisfies every inequality at once.
-> Shade each half-plane lightly, then keep the common part.
-> Example: y >= x and y <= 4.
-> The overlap is above y = x and below y = 4.
-> Point (1, 3) satisfies both: 3 >= 1 and 3 <= 4.
-> Point (5, 2) fails y >= x.
-> Example: y > x and y < x - 2 has no overlap.
-> Empty overlap means the system has no solution.
o Bounded Region
-> A bounded region fits inside a finite polygon.
-> Bounded describes extent, not whether edges are included.
-> A dashed boundary excludes its points, so it is not closed.
-> Example: x >= 0, y >= 0, x + y <= 6.
-> The vertices are (0, 0), (6, 0), and (0, 6).
-> Points on x + y = 6 are included by <=.
-> The region is a triangle with finite area.
-> Example with x > 0 excludes the y-axis edge.
-> It excludes (0, 0) and (0, 6), but stays bounded.
o Unbounded Region
-> An unbounded region extends without limit.
-> Example: y >= x + 1 and x >= 0.
-> The region continues up and to the right forever.
-> Its only vertex is (0, 1).
-> A single vertex does not make the region bounded.
-> Test points: (0, 2) and (10, 20) both satisfy.
-> Boundedness concerns the whole region, not one corner.
-> Parallel boundaries may leave a strip or no overlap.
-> Example: y > x and y < x + 2 gives an unbounded strip.
o Real Constraints
-> Count variables in whole items; add nonnegative constraints.
-> Example: x caps and y shirts, with x + y <= 40.
-> If at least 10 caps are needed, add x >= 10.
-> If shirts cost $8 and caps $5, $300 gives 5x + 8y <= 300.
-> The feasible region is the overlap of all constraints.
-> Check a candidate such as (20, 12): 20 + 12 <= 40.
-> Cost: 5(20) + 8(12) = 196 <= 300.
-> A fractional point may be infeasible if items are discrete.