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# help

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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

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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.

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(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.

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(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