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if x > y then x-a > y-a

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BukrsGmat
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if x > y then x-a > y-a [#permalink] New post   02 Jul 2012, 21:54
I do have a fundamental doubt ..help me in getting a sound understanding


if x > y then x-a > y-a
i) if a > 0
ii) if a y then x+a > y+a
i) if a > 0
ii) if a y then x*a > y*a
i) if a > 0
ii) if a y then \(\frac{x}{a} > \frac{y}{a}\)
i) if a > 0
ii) if a< 0
iii) if a= any value
iv) 0<a<1
v) -1<0<0
which one them true...
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Re: if x > y then x-a > y-a [#permalink] New post   02 Jul 2012, 23:17
Hi,

Highlighted the answers below:
sujit2k7 wrote:
I do have a fundamental doubt ..help me in getting a sound understanding
if x > y then x-a > y-a
i) if a > 0
ii) if a< 0
iii) if a= any value
iv) 0<a<1
v) -1<a<0



if x > y then x+a > y+a
i) if a > 0
ii) if a< 0
iii) if a= any value
iv) 0<a<1
v) -1<0<0

if x > y then x*a > y*a
i) if a > 0
ii) if a< 0
iii) if a= any value
iv) 0<a<1
v) -1<a<0

if x > y then \(\frac{x}{a} > \frac{y}{a}\)
i) if a > 0
ii) if a< 0
iii) if a= any value
iv) 0<a<1
v) -1<0<0
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cyberjadugar
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Re: if x > y then x-a > y-a [#permalink] New post   02 Jul 2012, 23:27
Hi,

Now explanations:
x > y (for example x,y = (4,3), (-1, -10) & so on...)

adding or subtracting any quantity on both sides will increase or decrease the value with same magnitude.
x+a>y+a
x-a>y-a

now, if a is multiplied on both sides;
let, x=4, y=3
if a>0, assuming a =1
then 4*1>3*1
if a<0 assuming a=-1,
then -4<-3, thus when a<0 the inequality sign changes.
similar is the case of division.

Regards,
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KarishmaB
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Re: if x > y then x-a > y-a [#permalink] New post   05 Jul 2012, 10:13
Expert Reply
sujit2k7 wrote:
I do have a fundamental doubt ..help me in getting a sound understanding


if x > y then x-a > y-a
i) if a > 0
ii) if a< 0
iii) if a= any value
iv) 0<a<1
v) -1<0<0
which one them true...


if x > y then x+a > y+a
i) if a > 0
ii) if a< 0
iii) if a= any value
iv) 0<a<1
v) -1<0<0
which one them true...



if x > y then x*a > y*a
i) if a > 0
ii) if a< 0
iii) if a= any value
iv) 0<a<1
v) -1<0<0
which one them true...



if x > y then \(\frac{x}{a} > \frac{y}{a}\)
i) if a > 0
ii) if a< 0
iii) if a= any value
iv) 0<a<1
v) -1<0<0
which one them true...



You can arrive at conclusions on your own.

Imagine x and y are on a number line.
x > y means x is to the right of y e.g.
_________y ________x __________

Now what happens when you add a positive number to both these numbers?
The numbers shift to the right on the number line.

_____________y+a__________x+a___________

Their relative placement is still the same. x+a > y+a

Similarly, when something is subtracted from both, their relative placement on the number line stays the same even though they both move to the left.

What happens when you multiply them by a positive number, say 2?

_________y__________x___________
becomes
________________ay_________________ax________

ax is still to the right of ay thought it is further to the right. So ax > ay

What happens when you divide by a positive number, say 2?
x/a and y/a come closer together but x/a is still to the right of y/a.
The inequality sign stays the same in all these cases.


What about the case when you multiply by a negative number, say -2?

_________y__________x___________
becomes
________________-ax_________________-ay________

-ax will have greater absolute value as compared to -ay but since it is negative, it will be smaller than -ay and will be to the left of -ay. Hence the inequality sign flip when you multiply/divide by a negative number.
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BukrsGmat
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Re: if x > y then x-a > y-a [#permalink] New post   05 Jul 2012, 10:24
Thnkx..both of u..let's take it little further

1. if 0< a<b<1.
if x < Y ( x, y any value )
then will x/a > y/b ?

2. -1>a>b>0
if x < Y ( x, y any value )
then will x/a > y/b ?
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KarishmaB
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Re: if x > y then x-a > y-a [#permalink] New post   05 Jul 2012, 22:15
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sujit2k7 wrote:
Thnkx..both of u..let's take it little further

1. if 0< a<b<1.
if x < Y ( x, y any value )
then will x/a > y/b ?

2. -1>a>b>0
if x < Y ( x, y any value )
then will x/a > y/b ?


If you are dealing with two inequalities, as in this case, all you can do is add the inequalities without worrying about the sign of the variables.

Say, if x < y and a < b
x+a < y + b

For anything else, you have to be mindful of the sign.

As for conditions given by you,

1. if 0< a<b<1.
if x < Y ( x, y any value )
then will x/a > y/b ?

Cannot Say!

Say x = 4, y = 8, a = 1/4, b = 1/2
Here x/a = y/b

If x = 4, y = 8, a = 1/4, b = 3/4
Here x/a > y/b

If x = 4, y = 8, a = 1/4, b = 5/16
Here x/a < y/b

Use the same logic for second one as well and find out.
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if x > y then x-a > y-a [#permalink] New post   05 Jul 2012, 23:09
Concerning your last question, I agree you can not say, but you can in a close case :
x > y > 0
a > b > 0

x/b > y/a (and not x/a and y/a as you said)

Proof :
x/b > y/b (because x>y, see before)
y/b > y/a (because b y/a
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if x > y then x-a > y-a [#permalink]
05 Jul 2012, 23:09
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