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# If a > b and c < d , which of the following MUST be true?

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If a > b and c < d , which of the following MUST be true?  [#permalink]

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10 Sep 2019, 21:38
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Question Stats:

55% (01:35) correct 45% (01:30) wrong based on 75 sessions

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Competition Mode Question

If $$a>b$$ and $$c<d$$, which of the following MUST be true?

A. $$a−d>c−b$$
B. $$a+d>b$$
C. $$b+c>a−d$$
D. $$b−d<a−c$$
E. $$a^2+d^2>b^2+c^2$$

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Re: If a > b and c < d , which of the following MUST be true?  [#permalink]

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10 Sep 2019, 22:35
1
given a > b and c < d

a-b > 0 and c-d < 0
from here we know that
a-b > c-d

a-c > b-d

b-d < a-c

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Re: If a > b and c < d , which of the following MUST be true?  [#permalink]

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10 Sep 2019, 22:41
1
Ans: D
Because, a>b , c<d. In inequalities, opposite signs can be subtracted. Or, c<d --> -c>-d . On adding a>b
+ (-c>-d)
= a-c>b-d
only d gives this answer choice
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Re: If a > b and c < d , which of the following MUST be true?  [#permalink]

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10 Sep 2019, 22:49
We are given a>b and c<d, we are to determine which of the answer choices must be true.
D is the answer. This is because a>b and c<d, hence b-d will always be less than a-c. We are subtracting a smaller number, c, from a, where a>b and subtracting a bigger number, d, where d>c from b. Definitely, a-c will always be greater than b-d.

In A, a-d>c-b, is not always true. when a=5, b=4, c=1, d=7, a-d(-2)>c-b(-3)
however if a=5, b=4, c=6, and d=7; a-d(-2)<c-b(3).

Similarly, in B, a+d>b is not always true. This is because, we don't know whether a,b, and d are greater than zero or otherwise. when a,b, and d are all greater than 0, then a+d>b. However, if a and b are positive and d is negative, it is possible than a+d<b. Hence B is not always true.

C b+c>a-d. This is also not always true. Depending on the magnitudes of a,b,c, and d, b+c>a-d. in other instances, b+c<a-d. To illustrate, let a=5, b=4, c=6 and d=7 then b+c(10)>a-d(-2). However, if a=5, b=4, c=-6 and d=-1, then b+c(-2)<a-d(6)

E a^2 + d^2 > b^2 + c^2. This is not always true. It is true for a,b,c, and d greater than zero. However, if a,b,c, and d are all negative, then it is not true.
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Re: If a > b and c < d , which of the following MUST be true?  [#permalink]

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10 Sep 2019, 23:48
1
If a>b and c<d, which of the following MUST be true?

A. a−d>c−b
B. a+d>b
C. b+c>a−d
D. b−d<a−c
E. $$a^2+d^2>b^2+c^2$$

a>b
d>c
a+d>b+c
b-d<a-c

IMO D
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Re: If a > b and c < d , which of the following MUST be true?  [#permalink]

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11 Sep 2019, 03:31
If a > b and c < d, which of the following MUST be true?

A. a−d > c−b
B. a+d > b
C. b+c > a−d
D. b−d < a−c
E. $$a^2 + d^2 > b^2 + c^2$$

Following cases are possible:

1. b < a < c < d
2. c < d < b < a
3. b < a = c < d
4. c < d = b < a
5. b < c < a = d
6. c < b < a = d
7. b = c < a < d
8. b = c < d < a

Attachment:
File comment: abcd

abcd.JPG [ 30.76 KiB | Viewed 820 times ]

Only case 1 and 2 are sufficient.

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Re: If a > b and c < d , which of the following MUST be true?  [#permalink]

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11 Sep 2019, 05:45
Quote:
If a>b and c<d, which of the following MUST be true?

A. a−d>c−b
B. a+d>b
C. b+c>a−d
D. b−d<a−c
E. $$a^2+d^2>b^2+c^2$$

given: a>b and c<d have different signs, we can subtract them, maintaining the former sign.
a>b-(c<d)=a-c>b-d…b-d<a-c

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Re: If a > b and c < d , which of the following MUST be true?  [#permalink]

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11 Sep 2019, 07:17
If a>b and c<d, which of the following MUST be true?
a>b --> a-b>0 -- (1)
c<d --> d>c --> d-c>0 --(2)
add (1) + (2) (now, we can add because both (1) & (2) have > 0 on rhs)
(a-b)+(d-c)>0
(a+d)-(b+c)>0
(a+d)>(b+c) --(3)

A. a−d>c−b
Incorrect, after rearranging, we can't get (3)
B. a+d>b
Incorrect, we can't get (3)
C. b+c>a−d
Incorrect, we can't get (3) after rearranging
D. b−d<a−c
Correct, after rearranging, (b+c)<(a+d) --> (a+d)>(b+c) which is (3)
E. (a^2 +d^2) >(b^2 +c^2)
Incorrect, we can't get (3) after rearranging
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Re: If a > b and c < d , which of the following MUST be true?  [#permalink]

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11 Sep 2019, 08:22
If a>b and c<d , which of the following MUST be true?

A. a−d>c−b
B. a+d>b
C. b+c>a−d
D. b−d<a−c
E. $$a^2$$+$$d^2$$>$$b^2$$+$$c^2$$

Eq1 : a > b
Eq2 : c<d ...Multiply both sides by -1, -c > -d
Add 1 & 2 - a-c > b-d

Scan the answer choices that match above expression

D it is.
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If a > b and c < d , which of the following MUST be true?  [#permalink]

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02 Nov 2019, 14:17
Bunuel wrote:

Competition Mode Question

If $$a>b$$ and $$c<d$$, which of the following MUST be true?

A. $$a−d>c−b$$
B. $$a+d>b$$
C. $$b+c>a−d$$
D. $$b−d<a−c$$
E. $$a^2+d^2>b^2+c^2$$

Nice question. Manipulate the answer choices to just be addition on both sides.

A) $$a−d>c−b$$ -------> a+b>c+d Clearly not necessarily true.

B) $$a+d>b$$ What if d is a high magnitude negative number?
Example:
a=1
b=0
d=-100
c=-101

1-100>0.... Not true

C) $$b+c>a−d$$ --------> b+c+d>a Clearly not necessarily true. Make a = a huge number

D) $$b−d<a−c$$ --------> b+c<a+d True. Both sides are pure addition and the larger numbers are on the same side.

E) $$a^2+d^2>b^2+c^2$$ Trap Answer. What if our small numbers are high magnitude negative numbers?
Example:
a=0
b=-10
d=0
c=-10

0+0>100+100

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If a > b and c < d , which of the following MUST be true?   [#permalink] 02 Nov 2019, 14:17
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