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二元一次不等式

當常數a,b不全為0時，\;不等式
\ax+by>c,ax+by<c,
\ax+by≥c,ax+by≤c稱為
\二元一次不等式，\;滿足不等式
\的實數對(x,y)稱為它的解。

>

<

≥

≤

+

-

二元一次不等式之圖解
\坐標平面上，\;將一個二元一次不等式的每一個解(x,y)視為
\一點，\;而以圖形標示所有解點所成的區域，\;即圖解該二元
\一次不等式。\;
\其中若x,y均為整數的點，\;稱為格子點。
\\;
\\;
\\;
\\;
\\;
\\;
\\;

如果要畫出幾個二元一次不等式
\聯立而成的不等式組的解區域時，
\那麼必須先描繪每一個二元一次
\不等式的解區域，\;再求其交集。

試求右圖所動應的聯立不等式組。

(0,8)

(0,5)

(6,2)

(8,0)

(10,0)

s = 0 (slider from 0 to max, step 1)
B = (15.04, -7.82)
poly_0, b_1, a_1, c_1, d_1 = Polygon(B, B + (18, 0), B + (18, 12), B + (0, 12))
C = PointIn(poly_0)
O = DynamicCoordinates(C, floor(x(C)), floor(y(C)))
u = Vector((x(B), y(O)), (x(B) + 18, y(O)))
v = Vector((x(O), y(B)), (x(O), y(B) + 12))
max = 37
A: Corner[4] - (0, 1.5)
a = 1 (slider from 1 to 5, step 1)
b = 2 (slider from 1 to 5, step 1)
c = 5 (slider from -5 to 5, step 1)
d = 3 (slider from 1 to 4, step 1)
e = 1 (slider from 1 to 2, step 1)
f_1 = If(e ≟ 1, (a * (x - x(O))) + (b * (y - y(O))) > c, If[e ≟ 2, (a * (x - x(O))) - (b * (y - y(O))) > c])
f_2 = If(e ≟ 1, (a * (x - x(O))) + (b * (y - y(O)))  <  c, If[e ≟ 2, (a * (x - x(O))) - (b * (y - y(O)))  <  c])
f_3 = If(e ≟ 1, (a * (x - x(O))) + (b * (y - y(O))) ≥ c, If[e ≟ 2, (a * (x - x(O))) - (b * (y - y(O))) ≥ c])
f_4 = If(e ≟ 1, (a * (x - x(O))) + (b * (y - y(O))) ≤ c, If[e ≟ 2, (a * (x - x(O))) - (b * (y - y(O))) ≤ c])
text2: "" + (LaTeX[If[a ≟ -1, "-", If[a ≟ 1, "", "" + a]]]) + "x" + (LaTeX[If[e ≟ 1, "+", "-"]]) + "" + (LaTeX[If[b ≟ -1, "-", If[b ≟ 1, "", "" + b]]]) + "y" + (LaTeX[If[d ≟ 1, ">", If[d ≟ 2, "<", If[d ≟ 3, "≥", "≤"]]]]) + "" + (LaTeX[c]) + ""
P_0 = PointIn(poly_0)
P = DynamicCoordinates(P_0, floor(x(P_0)), floor(y(P_0)))
textP: "P(" + (LaTeX[x(P) - x(O)]) + "," + (LaTeX[y(P) - y(O)]) + ")"
h: x(P) - x(O)
k: y(P) - y(O)
text5: "" + (LaTeX[a]) + "×" + (LaTeX[If[h  <  0, "(", ""]]) + "" + (LaTeX[h]) + "" + (LaTeX[If[h  <  0, ")", ""]]) + "" + (LaTeX[If[e ≟ 1, "+", "-"]]) + "" + (LaTeX[b]) + "×" + (LaTeX[If[k  <  0, "(", ""]]) + "" + (LaTeX[k]) + "" + (LaTeX[If[k  <  0, ")", ""]]) + "=" + (LaTeX[If[e ≟ 1, (a * h) + (b * k), (a * h) - (b * k)]]) + ""
g_1 = If(e ≟ 1, (a * (x - x(O))) + (b * (y - y(O))) = c, If[e ≟ 2, (a * (x - x(O))) - (b * (y - y(O))) = c])
g_2 = If(e ≟ 1, (a * (x - x(O))) + (b * (y - y(O))) = c, If[e ≟ 2, (a * (x - x(O))) - (b * (y - y(O))) = c])
L_{10}: x - x(O) + (2 * (y - y(O))) = 8
text10: "先畫出直線L:x+2y=8
\\\;
\\\;
\\" + (LaTeX[If[s ≥ 12, "故不等式的解為L與包含 \\P(" + h + " ," + k + " )點的半平面"]]) + ""
f_{12}(x, y) = x - x(O) + (2 * (y - y(O))) ≤ 8
L_{14}: (2 * (x - x(O))) - (y - y(O)) = 4
text9: "在座標平面上，\;畫出不等式
\\" + (LaTeX[If[s ≥ 13, "2x-y>4", "x+2y≤8"]]) + "的解區域。"
text14: "先畫出直線L:2x-y=4
\\\;
\\\;
\\" + (LaTeX[If[s ≥ 16, "故不等式的解為包含 \\P(" + h + " ," + k + " )點的半平面"]]) + ""
f_{16}(x, y) = (2 * (x - x(O))) - (y - y(O)) > 4
L_{19}: x - x(O) + y - y(O) = 4
L_{21}: x - x(O) + (2 * (y - y(O))) = 6
f_{20}(x, y) = x - x(O) + y - y(O) ≤ 4
f_{22}(x, y) = x - x(O) + (2 * (y - y(O))) ≤ 6
text19: "畫出直線L_1:x+y=4，\;" + (LaTeX[If[s ≥ 20, "再以原 \\點(0,0)代入找出x+y≤4的部分"]]) + "
\\" + (LaTeX[If[s ≥ 21, "畫出直線L_2:x+2y=6，\;"]]) + "" + (LaTeX[If[s ≥ 22, "再以原 \\點(0,0)代入找出x+2y≤6的部\\分；\;"]]) + "" + (LaTeX[If[s ≟ 23, "最後取交集，\;即為所求。"]]) + ""
Poly_{23}, b_2, f, g, i = Polygon(B, Intersect[x = x(B), L_{21}], O + (2, 2), Intersect[L_{19}, y = y(B)])
A_{241}: O + (0, 8)
A_{242}: O + (0, 5)
A_{243}: O + (8, 0)
A_{244}: O + (10, 0)
A_{245}: O + (6, 2)
L_{241}: (x - x(O)) / 8 + (y - y(O)) / 8 = 1
L_{242}: (x - x(O)) / 10 + (y - y(O)) / 5 = 1
Poly_{24}, j, l, m, n = Polygon(A_{245}, Intersect[x = x(B), L_{242}], B + (0, 12), Intersect[L_{241}, y = y(B) + 12])
text25: "先求通過(8,0),(0,8)的直線L_1
\\" + (LaTeX[If[s ≥ 26, "L_1:\frac{x}{8}+\frac{y}{8}=1"]]) + "" + (LaTeX[If[s ≥ 27, "⇒x+y=8"]]) + "
\\" + (LaTeX[If[s ≥ 28, "再求通過(10,0),(0,5)的直線L_2"]]) + "
\\" + (LaTeX[If[s ≥ 29, "L_2:\frac{x}{10}+\frac{y}{5}=1"]]) + "" + (LaTeX[If[s ≥ 30, "⇒x+2y=10"]]) + ""
text31: "將P(" + (LaTeX[h]) + "," + (LaTeX[k]) + ")代入x+y=8判斷
\\" + (LaTeX[If[s ≟ 33, "得x+y≤8"]]) + ""
f_{36}(x, y) = x - x(O) + (2 * (y - y(O))) ≥ 10
f_{33}(x, y) = x - x(O) + y - y(O) ≤ 8
list1: {"x+y≤4", "x+2y≤6"}
text18: "在坐標平面上，\;畫出一二元一次
\\不等式" + (LaTeX[TableText[{list1}, "v{"]]) + "的解區域。"
list2: {"x+y≤8", "x+2y≥10"}
text34: "將P(" + (LaTeX[h]) + "," + (LaTeX[k]) + ")代入x+2y=10判斷
\\" + (LaTeX[If[s ≥ 36, "得x+2y≥10"]]) + "
\\" + (LaTeX[If[s ≥ 37, TableText[{list2}, "v{"]]]) + ""
text7: "二元一次不等式的解區域" + (LaTeX[If[s ≟ 7, "\\在坐標平面上，二元一次不等式ax+by>c與ax+by<c \\的解區域分別是以直線L:ax+by=c為界線的兩側半平 \\面。在直線L外任取一點P(x_0,y_0)，若ax_0+by_0>c(或<c)，則 \\(1)\;點P所在的半平面就是不等式ax+by>c(或<c)的解 \\\;\;\;\;\;\;區域。 \\(2)\;不含點P所在的半平面就是不等式ax+by＜c(或＞c)的 \\\;\;\;\;\;\;解區域。"]]) + "" + (LaTeX[If[s ≟ 8, "\\描繪二元一次不等式的解區域時， \\(1)\;若不等式的解區域不含界線時，\;我們把界線畫為虛線；\; \\(2)\;若不等式的解區域包含界線時，\;我們把界線畫為實線。 \\\; \\\; \\\; \\\; \\\; \\\; \\\; \\\;\\\;\\\;\\\;"]]) + ""