Points to remember :

1. if x > y then (1/x) < (1/y)

( EDIT:Holds good only when BOTH x and Y share SAME sign - Thanks Ian)2. if x > y then -x < -y (that is the inequality gets reversed when both sides are multiplied by negative sign)

3. NEVER EVER , cross-multiply a variable (or) expression in an inequality blindly. You CAN cross multiply if and only if you are sure that the variable (or) expression being cross-multiplied is positive.

4. You can blindly cross multiply constant terms or numbers.

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For any real numbers, a; b,and c:

a < b is equivalent to a + c < b + c;

a > b is equivalent to a + c > b + c;

a = b is equivalent to a + c = b + c;

a >= b is equivalent to a + c >= b + c.

a <= b is equivalent to a + c <= b + c.

In other words, when we add or subtract the same number on both sides of an inequality, the direction of the inequality symbol is not changed.

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For any real numbers, a; b, and any positive number c:

a < b is equivalent to ac < bc;

a > b is equivalent to ac > bc.

a < b is equivalent to a/c < b/c;

a > b is equivalent to a/c > b/c.

For any real numbers, a; b, and any negative number c:

a < b is equivalent to ac > bc;

a > b is equivalent to ac < bc.

a < b is equivalent to a/c > b/c;

a > b is equivalent to a/c < b/c.

Similar statements hold for >= and <=

In other words, when we multiply or divide by a positive number on both sides of an inequality, the direction of the inequality symbol stays the same.

When we multiply or divide by a negative number on both sides of an inequality, the direction of the inequality symbol is reversed.

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|x| = x if x > 0

|x| = -x if x < 0

if |x| > y , then either x > y or -x > y

if |x| < y , then either x < y or -x < y

let "r" be a positive real number and "a" be a fixed real number, then

|x-a| < r implies a-r < x < a+r in other words x lies somewhere in between a-r and a+r

|x-a| > r implies x < a-r or x > a+r in other words, x lies outside a+r and a-r

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if k is a positive integer:

1 )if x^2 > k^2 and x> 0 then this implies x > k

2) k^x > 1 when x>0

3) 0< k^x < 1 when x < 0

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For word problems :

x is at least 30 implies x>=30 ( that is x is minimum 30)

x is at most 30 implies x<=30 ( that is x is maximum 30)

x cannot exceed 45 implies x <=45 ( that is x is maximum 45)

x must exceed 34 implies x > 34

x is between 7 and 12 implies 7 < x < 12

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Please add if i missed something.

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