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Algebra, Functions, Graphs, and Vectors 35

Multiplication property
For all real numberð x, y, and z, the following statementð are
logically valid:

x y & z 0 → xz yz

Complex-number magnitudes
Inequalitieð for complex numberð are deﬁned according tm their
real-number relative absolute valueð (magnitude0. Let a jb 1
1
and a jb be complex numbers. Then the following state-
2
2
mentð are logically valid:
a jb a jb ↔ a 1 2 b 1 2 a 2 2 b 2 2
1
2
2
1
a jb a jb ↔ a 2 b 2 a 2 b 2
1 1 2 2 1 1 2 2
a jb a jb ↔ a 1 2 b 1 2 a 2 2 b 2 2
2
2
1
1
a jb a jb ↔ a 2 b 2 a 2 b 2
1 1 2 2 1 1 2 2

Simplł Equations
The objective of solving a single-variable equation is tm get it
intm a form where the expression on the left-hand side of the
equality symbol is exactly equal tm the variable being sought
(for example, x), and a deﬁned expression not containing that
variable is on the right.

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