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Help With Add/Subtract/Mult/Divid Multiple Inequalities

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Help With Add/Subtract/Mult/Divid Multiple Inequalities [#permalink] New post 02 Jul 2013, 21:08
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Trying to remember the rules. Anyone know a good comprehensive place to find the rules?

I remember that you can add them so long as you keep the inequality sign the same way, but can't remember the other rules.
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Re: Help With Add/Subtract/Mult/Divid Multiple Inequalities [#permalink] New post 02 Jul 2013, 21:27
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amargius wrote:
Trying to remember the rules. Anyone know a good comprehensive place to find the rules?

I remember that you can add them so long as you keep the inequality sign the same way, but can't remember the other rules.


ADDING/SUBTRACTING INEQUALITIES:


You can only add inequalities when their signs are in the same direction:

If a>b and c>d (signs in same direction: > and >) --> a+c>b+d.
Example: 3<4 and 2<5 --> 3+2<4+5.

You can only apply subtraction when their signs are in the opposite directions:

If a>b and c<d (signs in opposite direction: > and <) --> a-c>b-d (take the sign of the inequality you subtract from).
Example: 3<4 and 5>1 --> 3-5<4-1.

RAISING INEQUALITIES TO EVEN/ODD POWER:


A. We can raise both parts of an inequality to an even power if we know that both parts of an inequality are non-negative (the same for taking an even root of both sides of an inequality).
For example:
2<4 --> we can square both sides and write: 2^2<4^2;
0\leq{x}<{y} --> we can square both sides and write: x^2<y^2;

But if either of side is negative then raising to even power doesn't always work.
For example: 1>-2 if we square we'll get 1>4 which is not right. So if given that x>y then we can not square both sides and write x^2>y^2 if we are not certain that both x and y are non-negative.

B. We can always raise both parts of an inequality to an odd power (the same for taking an odd root of both sides of an inequality).
For example:
-2<-1 --> we can raise both sides to third power and write: -2^3=-8<-1=-1^3 or -5<1 --> -5^2=-125<1=1^3;
x<y --> we can raise both sides to third power and write: x^3<y^3.

THEORY ON INEQUALITIES:


x2-4x-94661.html#p731476
inequalities-trick-91482.html
data-suff-inequalities-109078.html
range-for-variable-x-in-a-given-inequality-109468.html
everything-is-less-than-zero-108884.html
graphic-approach-to-problems-with-inequalities-68037.html
inequations-inequalities-part-154664.html
inequations-inequalities-part-154738.html

QUESTIONS:


All DS Inequalities Problems to practice: search.php?search_id=tag&tag_id=184
All PS Inequalities Problems to practice: search.php?search_id=tag&tag_id=189

700+ Inequalities problems: inequality-and-absolute-value-questions-from-my-collection-86939.html

Hope it helps.
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Multiplying Two Inequalities [#permalink] New post 03 Jul 2013, 14:19
Hi,

Is there any general rule for multiplication or division of two inequalities?

If
a<x<b and c<y<d
then
a+c<x+y<b+d
a-d<x-y<b-c

Now is there any rule to find the relationships for xy or x/y?

Thank you in advance!
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Re: Multiplying Two Inequalities [#permalink] New post 03 Jul 2013, 19:39
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alphabeta1234 wrote:
Hi,

Is there any general rule for multiplication or division of two inequalities?

If
a<x<b and c<y<d
then
a+c<x+y<b+d
a-d<x-y<b-c

Now is there any rule to find the relationships for xy or x/y?

Thank you in advance!



Multiplication - Yes but very constrained.
If both sides of both inequalities are positive and the inequalities have the same sign, you can multiply them.

x < a
y < b
xy < ab
Given x, y, a, b are all positive.

Otherwise
-2 < -1
10 < 30
Multiply: -20 < -30 (Not correct)
or
-2 < 7
-8 < 1
Multiply: 16 < 7 (Not correct)

For division, this may not hold.
e.g
2 < 10
4 < 40
Divide: 1/2 < 1/4

But if both sides of both the inequalities are positive and the signs of the inequality are opposite, then you can divide them

x < a
y > b

x/y < a/b (given all x, y, a, b are positive)
The final inequality takes the sign of the numerator. Take examples.
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Re: Help With Add/Subtract/Mult/Divid Multiple Inequalities [#permalink] New post 03 Jul 2013, 22:24
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Re: Help With Add/Subtract/Mult/Divid Multiple Inequalities [#permalink] New post 31 Aug 2013, 05:25
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Re: Help With Add/Subtract/Mult/Divid Multiple Inequalities [#permalink] New post 16 Sep 2014, 15:29
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Re: Help With Add/Subtract/Mult/Divid Multiple Inequalities   [#permalink] 16 Sep 2014, 15:29
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