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