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Let \(a,b,c\) and \(d\) be real numbers such that \(a > b\) and \(c > d.\) Which of the statements below must be true. (A) \(a + c > b + d.\) (B) \(2a + 3c > 2b + 3d.\) (C) \(a - c > b - d.\) (D) \(ac > bd.\) (E) \(a^2 + c^2 > b^2 + d^2.\) (F) \(a^3 + c^3 > b^3 + d^3.\)

 May 20, 2019
 #1
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Let  a,  b,  c,  and  d  be real numbers such that  a > b  and  c > d .

\(\text{__________________________________________________________________________________}\)

 

(A)     a + c  >  b + d

 

We know.....

 

a  >  b_____and_____c  >  d

 

Add  d  to both sides of the first inequality.

 

a + d  >  b + d

 

If we substitute  c  for  d  on the left side it will make the left side bigger,

and the inequality will still be true. So it is always true that

 

a + c  >  b + d

\(\text{__________________________________________________________________________________}\)

 

(B)     2a + 3c  >  2b + 3d

 

a  >  b _____and_____ c  >  d
2a  >  2b and 3c  >  3d

 

By the same logic from part (A), we can conclude that it is also true that     2a + 3c  >  2b + 3d

\(\text{__________________________________________________________________________________}\)

 

(C)     a - c  >  b - d


Counterexample:     Let   a = 2 ,  b = 1 ,  c = 2 ,  d = 1

 

a  >  b    and    c  >  d    but it is not true that    a - c  >  b - d    because    2 - 2  >  1 - 1    is not true.

\(\text{__________________________________________________________________________________}\)

 

(D)     ac  >  bd

 

Counterexample:     Let   a = 5 ,  b = 1 ,  c = -1 ,  d = -2

 

a  >  b    and    c  >  d    but it is not true that    ac  >  bd    because    (5)(-1)  >  (1)(-2)    is not true.

\(\text{__________________________________________________________________________________}\)

 

(E)     a2 + c2  >  b2 + d2

 

Counterexample:     Let   a = 2 ,  b = -5 ,  c = 2 ,  d = 1

 

a  >  b    and    c  >  d    but it is not true that    a2 + c2  >  b2 + d2    because    22 + 22  >  (-5)2 + 12    is not true.

\(\text{__________________________________________________________________________________}\)

 

(F)     a3 + c3  >  b3 + d3

 

If     a  >  b     then     | a3 |  >  | b3 |     and     a3  >  b3

 

If     c  >  d     then     | c3 |  >  | d3 |     and     c3  >  d3

 

By the same logic from part  (A) , we can conclude that it is also true that     a3 + c3  >  b3 + d3

 May 20, 2019

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