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第 15 週 · 不等式與線性規劃 · 第 4 節

二元一次不等式的圖像(半平面)

前面在數線上解不等式;現在升到平面 —— 二元一次不等式的解是「一整片區域」(半平面)。先畫界線,再用測試點決定取哪一半。

▶ 第一關: 半平面 —— 一條直線分平面成兩半
y = mx + cupper halfy > mx + clower halfy < mx + cone line splits the plane into two half-planes
Oy = x + 1shade: y > x + 1(upper half)(0, 5) ✓y > mx + c : shade ABOVE the boundary
▶ 第二關: 取哪一半 —— 測試點(原點)代入
boundary linetest point (0,0)plug in →holds? shade THIS sidefails? shade the OTHER sidepick a test point, substitute, decide the half
O2x + y = 4(0,0): 0 ≤ 4 ✓shade side WITH origin(0,4)(2,0)
Ox + y = 5(0,0): 0 > 5 ✗shade OTHER side(not the origin side)strict > : dashed line + take the half away from origin
▶ 第三關: 虛線 / 實線 + 化 y 形式(除負翻號)
> or <dashed (boundary NOT included)≥ or ≤solid (boundary IS included)
−y < 2×(−1) , flip! → y > −2was < (lower) ... now > (upper) — half SWAPS
x = 2x ≥ 2(right side)y = 3y ≤ 3 (below)
⚔ BOSS: 判斷某點在不在陰影 + 綜合
y = x (dashed)shade y > xin ✓on line: NOT in (strict)plug a point into the ORIGINAL inequality to test