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) \(a^3 + c^3 > b^3 + d^3.\)
+9466 Let a, b, c, and d be real numbers such that a > b and c > d .
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(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
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(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
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(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.
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(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
