Page 109 - Elementary Algebra Exercise Book I

ELEMENTARY ALGEBRA EXERCISE BOOK I inequAlities

ab c

3.66 a, b, c are real numbers, and a + b + c< 0, show b ca ≥ 0.

ca b

2
2
2
2
Proof: a + b ≥ 2ab, b + c ≥ 2bc, c + a ≥ 2ca , and add them up to obtain
2
2
2 2
2 2
2(a + b + c ) ≥ 2(ab + bc + ca), thus (ab + bc + ca) − (a + b + c ) ≤ 0.
2 2
2(a + b + c ) ≥ 2(ab + bc + ca) 2 2 2

1

ab c
ab c a + b + cb c 1 bc bc 1 b 1 b c c

a + b + cb c

ab c a + b + cb c 1 bc 1 b c

1
b

1

ab
b ca =
=(a+b+c) 1
=
b ca = a + b + cca =(a+b+c) 1 ca ca =(a+b+c) 0 c − ba = − c c
=(a+b+c) 0 c − ba − c

c a + b + cca bc

a + b + cb c

=

b ca a + b + cca =(a+b+c) 1 ca =(a+b+c) 0 c − ba − c =
b

1 ab ab
a + b + ca b b
ca b a + b + ca =(a+ 1 0 a − bb − c bb − c =

ca bca a + b + cca b+c) 1 ca =(a+b+c) 0 c − ba − c

=

2 0 a − bb − c
ca b a + b + ca b 1 ab 0 a −

1 ab
2
(a + b + c)[−(b − c) − (a − b)(a − c)] = (a + b + c)[(ab + bc + ca) − (a + b + c )] ≥ 0 2 0 a − bb − c
2

a + b + ca b
ca b
2
2
2
(a + b + c)[−(b − c) − (a − b)(a − c)] = (a + b + c)[(ab + bc + ca) − (a + b + c )] ≥ 0
2

2 − (a − b)(a − c)] = (a + b + c)[(ab + bc + ca) − (a + b + c )] ≥ 0.
2
2
2
2
(a + b + c)[−(b − c)
2
2
2
(a + b + c)[−(b − c) − (a − b)(a − c)] = (a + b + c)[(ab + bc + ca) − (a + b + c )] ≥ 0
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