Select the correct alternative from the given choices. Which...

Select the correct alternative from the given choices.
Which of the following is true?
(A) $|x+y| \leq|x|+|y|$
(B) $\left|\frac{x}{y}\right|=\frac{|x|}{|y|}, y \neq 0$
(C) $|x-y| \geq|| x|-| y||$
(D) All the above

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Key Concepts

Quotient Property of Absolute Values

This property describes how absolute values behave under division, stating that when dividing one non-zero number by another, the absolute value of the quotient is equal to the quotient of their absolute values. It is useful in simplifying expressions and solving equations involving fractions.

Reverse Triangle Inequality

The reverse triangle inequality establishes a lower bound for the absolute value of the difference between two numbers, asserting that it is at least as large as the absolute difference of their absolute values. This property is important for understanding the extent to which two quantities can differ in magnitude.

Absolute Value

Absolute value is a fundamental concept in mathematics that measures the non-negative magnitude of a real number, disregarding its sign. It provides the basis for many operations and properties in algebra and analysis.

Triangle Inequality

The triangle inequality is a key property of absolute values which states that for any two real numbers, the absolute value of their sum is less than or equal to the sum of their absolute values. This property illustrates how distances or magnitudes add together in a way analogous to the sides of a triangle.

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