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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:

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

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

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

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

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

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

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

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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:

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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:

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

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