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Here is a list of some important properties of inequalities:
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).
While if x and y have different signs, then even taking the reciprocal would not change the sign of the inequality.
(A substitute of taking the reciprocal is cross-multiplication, where you have to be careful of the sign of the number you are transporting.)
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|.
6. If |x - a| < b, then (a-b) < x < (a+b)
7. If |x - a| > b, then: x <(a-b) or x> (a+b)
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).
While if x and y have different signs, then even taking the reciprocal would not change the sign of the inequality.
(A substitute of taking the reciprocal is cross-multiplication, where you have to be careful of the sign of the number you are transporting.)
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|.
6. If |x - a| < b, then (a-b) < x < (a+b)
7. If |x - a| > b, then: x <(a-b) or x> (a+b)
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