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第 15 週 · 不等式與線性規劃 · 第 5 節

聯立二元一次不等式 · 可行域

上節一個不等式畫出一個「半平面」；這節把幾個不等式聯立，它們的公共重疊區就是「可行域」。用「重疊＝交集」把它講透，再學求頂點。

▶ 第一關： 半平面 ＋ 判斷點在不在
boundary: x + y = 4O (0,0)x + y ≤ 4(shaded side)test O: 0+0=0≤4 ✓so shade O's side
(0,0)... in : 0≤4 ✓(3,3) out : 6≤4 ✗plug the point in: TRUE = inside, FALSE = outside
point (2, 1) -- check ALL three:x ≥ 0 : 2 ≥ 0 ✓y ≥ 0 : 1 ≥ 0 ✓x + y ≤ 4 : 3 ≤ 4 ✓all ✓ => inside feasible region
▶ 第二關： 可行域 ＝ 半平面的公共重疊
x ≥ 0: right of y-axisy ≥ 0: above x-axisoverlap of all three= feasible regionmore layers overlap = deeper colour = the region we want
(0,0)(4,0)(0,4)x≥0 ∩ y≥0∩ x+y≤4= triangle
feasible region(a polygon)dots = vertices(line crossings)edges are boundary segments; corners are where two boundaries cross
▶ 第三關： 求頂點（聯立界線）
x + y = 4y = 0(4, 0)solve together:y=0 into x+y=4=> x = 4
x + y = 4(0,0)(4,0)(0,4)feasibleregionx ≥ 0, y ≥ 0, x + y ≤ 4 -- the classic triangle, 3 vertices
⚔ BOSS： 可行域頂點 ＋ 判點 ＋ 形狀綜合
x = 5y = 4(0,0)(5,0)(5,4)(0,4)bounded rectanglex≥0, y≥0, x≤5, y≤4 -- 4 vertices, bounded
BOUNDEDclosed -- you can box it inUNBOUNDEDextends forever →no upper bound -- open