4) (a) Show that if a 0 and b 2 0, then |a + b| = |a| + |b|.(b) Show that if a(c) Show that if a < 0 and b 2 0, then |a b| < \a| + |b|.(d) Show that if a < 0 and b(e) Explain why the previous four items imply that0, then |a + b| < \a| + |b|..0 and b0, then lab= |a| + |b|.labllal beal numbers a and bfor all

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Asked Sep 18, 2019
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Hello, I am confused on how to prove this question.

4) (a) Show that if a 0 and b 2 0, then |a + b| = |a| + |b|.
(b) Show that if a
(c) Show that if a < 0 and b 2 0, then |a b| < \a| + |b|.
(d) Show that if a < 0 and b
(e) Explain why the previous four items imply that
0, then |a + b| < \a| + |b|..
0 and b
0, then lab= |a| + |b|.
labllal b
eal numbers a and b
for all
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4) (a) Show that if a 0 and b 2 0, then |a + b| = |a| + |b|. (b) Show that if a (c) Show that if a < 0 and b 2 0, then |a b| < \a| + |b|. (d) Show that if a < 0 and b (e) Explain why the previous four items imply that 0, then |a + b| < \a| + |b|.. 0 and b 0, then lab= |a| + |b|. labllal b eal numbers a and b for all

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Expert Answer

Step 1

(a)By definition, we have

ifx 20
x
H-
-r ifx<0
so that xx2 for allxe R
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ifx 20 x H- -r ifx<0 so that xx2 for allxe R

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Step 2

Now,

a b(a+b)
Va b +2]a|b| (a2a and bb)
<laf +Ibf + 2la|b| (a5.r=kf)
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a b(a+b) Va b +2]a|b| (a2a and bb) <laf +Ibf + 2la|b| (a5.r=kf)

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Step 3

After solvi...

a+bl=\{Ia|+l6|) *
(la= a and b=b if a 20 and b 2 0
lal+bl0
= la+bl
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a+bl=\{Ia|+l6|) * (la= a and b=b if a 20 and b 2 0 lal+bl0 = la+bl

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