Absolute minimum

Algebra Level 2

As $x$ and $y$ range over all non-zero real values, what is the minimum value of $\frac { |x+y| } { | x | + | y | }?$

Details and assumptions

The notation $| \cdot |$ denotes the absolute value. The function is given by $|x | = \begin{cases} x & x \geq 0 \\ -x & x < 0 \\ \end{cases}$ For example, $|3| = 3, |-2| = 2$ .

The answer is 0.

1 solution

Arron Kau Staff
May 13, 2014

Since the numerator of the fraction is non-negative, and the denominator of the fraction is strictly positive, the value of the fraction is non-negative.

Since $\frac{ | -1 + 1 | } { |-1| + |1| } = \frac { 0} { 2} = 0$ , thus we know that the minimum value is 0.

Note: The triangle inequality states that $|x + y | \leq |x| + |y|$ , so we know that the expression is at most 1.

Absolute minimum

The answer is 0.

1 solution

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