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12 Jan 2020, 07:56
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Here are the some properties of the Inequalities.
1) If both the sides of an inequality as L.H.S and R.H.S are Positive, then you can square it.
2) If two inequalities are in same direction such as x>0,y>0 you can add them. If both are in different direction such as x>0,y<0 you can subtract one from another.
3) If two inequalities are in different direction such as x+y>0,y-x<0 then you can multiply one of them with -1 to change the sign. (example,y-x<0, if you multiply with -1 then x-y>0)
Kindly add if you know other properties of Inequalities below.
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1) If both the sides of an inequality as L.H.S and R.H.S are Positive, then you can square it.
2) If two inequalities are in same direction such as x>0,y>0 you can add them. If both are in different direction such as x>0,y<0 you can subtract one from another.
3) If two inequalities are in different direction such as x+y>0,y-x<0 then you can multiply one of them with -1 to change the sign. (example,y-x<0, if you multiply with -1 then x-y>0)
Kindly add if you know other properties of Inequalities below.
Kudos if you like my post!
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13 Jan 2020, 08:57
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Important property: Do not cross multiply the variables in the inequality equation unless one knows the +ve and -ve sign of the variables.
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16 Jan 2020, 00:44
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Here is a list of more properties:
1. ADDITION/SUBTRACTION
As long as you are adding/subtracting quantities in an inequality, its sign remains the same.
Eg: 2x + 5 < 3: When you take the 5 to the RHS and subtract, the sign will stay the same. You will get: 2x < -2.
2. MULTIPLICATION/DIVISION
If you multiply/divide by a positive number, the sign of the inequality does not change.
If you multiply/divide by a negative number, the sign of the inequality flips.
Eg: 2x < -2 => x < -1
while, -2x < -2 => x > (-2/-2) = 1
3. RECIPROCAL
If x < y, then 1/x > 1/y if and only if x and y have the same sign (both positive or both negative).
4. MINIMUM/MAXIMUM VALUES
Consider this example:
If -1<x<5 and -10<y<-2,
then to find range of expressions made of x and y, eg, x+y, xy, x-y, 2x+y, etc.,
we should evaluate these expressions using all four extreme cases.
So, for xy, we should find this product 4 times: (-1 x -10), (-1 x -2), (5 x -10), (5 x -2)= 10, 2, -50, -10.
Hence, range of xy is : -50 < xy < 10.
5. If \(x^{2}\) < \(y^{2}\), then |x| < |y|.
1. ADDITION/SUBTRACTION
As long as you are adding/subtracting quantities in an inequality, its sign remains the same.
Eg: 2x + 5 < 3: When you take the 5 to the RHS and subtract, the sign will stay the same. You will get: 2x < -2.
2. MULTIPLICATION/DIVISION
If you multiply/divide by a positive number, the sign of the inequality does not change.
If you multiply/divide by a negative number, the sign of the inequality flips.
Eg: 2x < -2 => x < -1
while, -2x < -2 => x > (-2/-2) = 1
3. RECIPROCAL
If x < y, then 1/x > 1/y if and only if x and y have the same sign (both positive or both negative).
4. MINIMUM/MAXIMUM VALUES
Consider this example:
If -1<x<5 and -10<y<-2,
then to find range of expressions made of x and y, eg, x+y, xy, x-y, 2x+y, etc.,
we should evaluate these expressions using all four extreme cases.
So, for xy, we should find this product 4 times: (-1 x -10), (-1 x -2), (5 x -10), (5 x -2)= 10, 2, -50, -10.
Hence, range of xy is : -50 < xy < 10.
5. If \(x^{2}\) < \(y^{2}\), then |x| < |y|.
