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24 Oct 2006, 07:57
Kudos
Bookmarks
Hi All,
Of late, I have seen so many inequality problems. I am a bit confused about the properties related to inequalities.
If someone can briefly explain the same in this thread, it will b of gr8 help.
Basically, I am looking @ stuff like, when can we cross multiply terms on either side of the inequality, when does the sign change, what props of
inequalities are different from those of equality n stuff like that.
Yezz, Anand, Yogesh.....Throw some light here
If neyone who responds to this thread can come up with examples, it will b easier to understand
Thanks in Advance
girikorat
Of late, I have seen so many inequality problems. I am a bit confused about the properties related to inequalities.
If someone can briefly explain the same in this thread, it will b of gr8 help.
Basically, I am looking @ stuff like, when can we cross multiply terms on either side of the inequality, when does the sign change, what props of
inequalities are different from those of equality n stuff like that.
Yezz, Anand, Yogesh.....Throw some light here
If neyone who responds to this thread can come up with examples, it will b easier to understand
Thanks in Advance
girikorat
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24 Oct 2006, 12:00
Kudos
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Basic Rules for Inequalities:
(in the example: a>b>0, c>d>0)
You need to flip signs when both side are multiplied by a negative number:
-a<-b, -c<-d
You need to flip signs when 1 is divided by both side:
1/a<1/b, 1/c<1/d
But if you x<0 you cant say that 1/x >0 as 1/0 is not defined.
You can only add or multiply them when their signs are in the same direction:
a+c>b+d
ac>bd
You can only apply substractions and divisions when their signs are in the opposite directions:
a>b, d<c
a-d>b-c
a/d>b/c
(You can't say a/c>b/c. It is WRONG)
Deal with negative numbers:
-a<-b<0, -c<-d<0
Then
-a-c<-b-d<0
-a-(-d)<-b-(-c)
However the sign needs to be flipped one more time if you are doing multiplication or division (because you are multiplying/dividing a negative number):
(-a)*(-c)>(-b)*(-d)
(-a)/(-d)>(-b)/(-c)
For example:
If x<-4, y<-2, we know that xy>8, but we don't know how x/y compare to (-4)/(-2)=2 since you can only do division when their signs are in different directions
If x>-4 and y<-2 then x/y<2 but we don't know how xy is compared to 8 since we can only do multiplication when their signs are the same direction.
It is easier to do the derivation, though, if you first change them to postive. For example:
If x<-4, y<-2, then -x>4, -y>2, xy>8
If x<-4, y<2, then -x>4, y<2, -x/y>2, x/y<-2
Cancelling out "Common Terms" on Both Sides of an Equation
You need to be very careful when you do algebra derivations. One of the common mistakes is to divide both side by "a common term". Remember you can only do this safely if the "common term" is a constant. However you CAN't do it if it contains a variable.
Example:
x(x-2)=x
You can't cancel out the x on both side and say x=3 is the solution. You must move the x on the right side to the left side.
x(x-2)-x=0
x(x-2-1)=0
The solutions are: x=0 and x=3
The reason why you can't divided both sides by x is that when x is zero, you can't divide anything by zero.
Equally important if not more, is that you CAN'T multiple or divide a "common term" that includes a variable from both side of an inequality. Not only it could be zero, but it could also be negative in which case you would need to flip the sign.
Example:
x^2>x
You CAN'T divided both sides by x and say x>1. What you have to do is to move the right side to the left:
x^2-x>0
x(x-1)>0
Solution would be either both x and x-1 are greater than zero, or both x and x-1 are smaller than zero. So your solution is: x>1 or x<0
Example:
x>1/x
Again you CAN'T multiply both sides by x because you don't know if x is positive or negative. What you have to do is to move the right side to the left:
x-1/x>0
(x^2-1)/x>0
If x>0 then x^2-1>0 =>x>1
If x<0 then x^2-1<0 =>x>-1
Therefore your solution is x>1 or 0>x>-1.
You could also break the original question to two branches from the beginning:
x>1/x
if x>0 then x^2>1 =>x>1
if x<0 then x^2<1 => x>-1
Therefore your solution is x>1 or 0>x>-1.
Hope That Helps
Yogesh
(in the example: a>b>0, c>d>0)
You need to flip signs when both side are multiplied by a negative number:
-a<-b, -c<-d
You need to flip signs when 1 is divided by both side:
1/a<1/b, 1/c<1/d
But if you x<0 you cant say that 1/x >0 as 1/0 is not defined.
You can only add or multiply them when their signs are in the same direction:
a+c>b+d
ac>bd
You can only apply substractions and divisions when their signs are in the opposite directions:
a>b, d<c
a-d>b-c
a/d>b/c
(You can't say a/c>b/c. It is WRONG)
Deal with negative numbers:
-a<-b<0, -c<-d<0
Then
-a-c<-b-d<0
-a-(-d)<-b-(-c)
However the sign needs to be flipped one more time if you are doing multiplication or division (because you are multiplying/dividing a negative number):
(-a)*(-c)>(-b)*(-d)
(-a)/(-d)>(-b)/(-c)
For example:
If x<-4, y<-2, we know that xy>8, but we don't know how x/y compare to (-4)/(-2)=2 since you can only do division when their signs are in different directions
If x>-4 and y<-2 then x/y<2 but we don't know how xy is compared to 8 since we can only do multiplication when their signs are the same direction.
It is easier to do the derivation, though, if you first change them to postive. For example:
If x<-4, y<-2, then -x>4, -y>2, xy>8
If x<-4, y<2, then -x>4, y<2, -x/y>2, x/y<-2
Cancelling out "Common Terms" on Both Sides of an Equation
You need to be very careful when you do algebra derivations. One of the common mistakes is to divide both side by "a common term". Remember you can only do this safely if the "common term" is a constant. However you CAN't do it if it contains a variable.
Example:
x(x-2)=x
You can't cancel out the x on both side and say x=3 is the solution. You must move the x on the right side to the left side.
x(x-2)-x=0
x(x-2-1)=0
The solutions are: x=0 and x=3
The reason why you can't divided both sides by x is that when x is zero, you can't divide anything by zero.
Equally important if not more, is that you CAN'T multiple or divide a "common term" that includes a variable from both side of an inequality. Not only it could be zero, but it could also be negative in which case you would need to flip the sign.
Example:
x^2>x
You CAN'T divided both sides by x and say x>1. What you have to do is to move the right side to the left:
x^2-x>0
x(x-1)>0
Solution would be either both x and x-1 are greater than zero, or both x and x-1 are smaller than zero. So your solution is: x>1 or x<0
Example:
x>1/x
Again you CAN'T multiply both sides by x because you don't know if x is positive or negative. What you have to do is to move the right side to the left:
x-1/x>0
(x^2-1)/x>0
If x>0 then x^2-1>0 =>x>1
If x<0 then x^2-1<0 =>x>-1
Therefore your solution is x>1 or 0>x>-1.
You could also break the original question to two branches from the beginning:
x>1/x
if x>0 then x^2>1 =>x>1
if x<0 then x^2<1 => x>-1
Therefore your solution is x>1 or 0>x>-1.
Hope That Helps
Yogesh
24 Oct 2006, 12:03
Kudos
Bookmarks
Yezz and Anand kindly add some more inputs to this thread..lets collect all the inequality properties if possible.. 
