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This essential trains for: AMC-10, AMC-12, AIME, Math Kangaroo 9-10.

Minimum value of an absolute value is zero:

The absolute value of a number is zero if and only if the number is zero:

The absolute value of x is an even function (not so translations of it!):

The absolute value of the product is equal to the product of the absolute values:

The absolute value of the quotient is equal to the quotient of the absolute values:

The triangle inequality. The absolute value of the sum is generally smaller than the sum of the absolute values!

The absolute value of the difference is generally larger than the difference of the absolute values:

The absolute value of the difference is generally larger than the absolute value of the difference of the absolute values:

For any positive constant, the absolute value of a number is smaller than the constant if and only if the number is larger than the opposite of the constant and smaller than the constant:

For any positive constant, the absolute value of a number is larger than the constant if and only if the number is smaller than the opposite of the constant and larger than the constant:

The value of a number is always larger than the opposite of its absolute value and smaller than its absolute value: