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Kernul
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Homework Statement
Prove the following facts about inequalities. [In each problem you will have to consider several
cases separately, e.g. ##a > 0## and ##a = 0##.]
(a) If ##a \leq b##, then ##a + c \leq b + c##.
(b) If ##a \geq b##, then ##a + c \geq b + c##.
(c) If ##a \leq b## and ##c \geq 0##, then ##ac \leq bc##.
(d) If ##a \leq b## and ##c \leq 0##, then ##ac \geq bc##.
Homework Equations
The Attempt at a Solution
So, I've tried to prove the first one (the second is basically the first one but with inequalities inverted) the following way:
If ##a > 0, b > 0## or ##a < 0, b < 0## (do I really have to say that ##b> 0## too?)
##a \leq b \implies a + c \leq b + c##
##a + (c + (-c)) \leq b + (c + (-c))##
##(a + c) + (-c) \leq (b + c) + (-c)##
##a + c \leq b + c##
If ##a = 0, b > 0##
##0 \leq b \implies c \leq b + c##
##(c + (-c)) \leq b + (c + (-c))##
##c + (-c) \leq (b + c) + (-c)##
##c \leq b + c##
So it's proved, right?
Now, going to (c), we have this ##c \geq 0## that stops me from using ##c^{-1}## to prove them because it's not ##c > 0##.
Which other way can I prove the last two?
EDIT: Sorry, I didn't notice the inequalities were wrong.
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