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ax + by + c > 0,

ax + by + c > 0,

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.3: Lines
Problem 32E
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You are now allowed to assume that the half-planes determined by the line with the equation ax+by +
c = 0 correspond to the points (x, y) so that ax + by + c < 0 and ax + by + c > 0, respectively. Using
this, show that axiom B4(i) holds. (Hint. Suppose (q, r) and (s, t) are on the same side of the given line
and that (s, t) and (u, v) are on the same side of the given line. ‹en construct the parametrized line
through (q, r) and (u, v). Consider the mapping
λ γ
7
→ a(q − qλ + uλ) + b(r − rλ + vλ) + c and note that it is continuous and either increasing or decreasing. Use this fact to show that, for every
λ, γ(λ) > 0 or γ(λ) < 0, depending on which half-plane the points are on.)

You are now allowed to assume that the half-planes determined by the line with the equation ax+by+ c = 0 correspond to the points (x, y) so that ax+by+c<0 and ax+by+c>0, respectively. Using this, show that axiom B4(i) holds. (Hint. Suppose (q, r) and (s, t) are on the same side of the given line and that (s, t) and (u, v) are on the same side of the given line. Then construct the parametrized line through (q, r) and (u, v). Consider the mapping Xa(q-qλ + uλ) + b(r = rλ + vλ) + c
Transcribed Image Text:You are now allowed to assume that the half-planes determined by the line with the equation ax+by+ c = 0 correspond to the points (x, y) so that ax+by+c<0 and ax+by+c>0, respectively. Using this, show that axiom B4(i) holds. (Hint. Suppose (q, r) and (s, t) are on the same side of the given line and that (s, t) and (u, v) are on the same side of the given line. Then construct the parametrized line through (q, r) and (u, v). Consider the mapping Xa(q-qλ + uλ) + b(r = rλ + vλ) + c
and note that it is continuous and either increasing or decreasing. Use this fact to show that, for every A, y(x) > 0 or y(x) < 0, depending on which half-plane the points are on.)
Transcribed Image Text:and note that it is continuous and either increasing or decreasing. Use this fact to show that, for every A, y(x) > 0 or y(x) < 0, depending on which half-plane the points are on.)
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