We use cookies to personalise content and advertisements and to analyse access to our website. Furthermore, our partners for online advertising receive pseudonymised information about your use of our website. cookie policy and privacy policy.
 
+0  
 
0
90
1
avatar

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
avatar+8759 
+3

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

30 Online Users

avatar
avatar
avatar
avatar
avatar