Is the absolute value function a linear function? I'm inclined to say yes, as it doesn't involve exponentiation, roots, logarithmic or trigonometric functions, but I watched a video where the teacher said that the absolute value function is "clearly non-linear". Why would he say that? Is he wrong? 
Wikipedia's graph for abs:

 A: A function $f(x)$ is linear if it satisfies the property
$$f(ax+y) = af(x) + f(y).$$
Let's try $a=-1$, $x=1$, $y=0$:
$$\begin{align*}
|ax+y| = |-1| &= 1\\
-1\ |1|+|0| &= -1\end{align*},$$
so $f(x) = |x|$ is not linear.
Sometimes (especially in geometry) "linear" is understood to mean affine. A function $f(x)$ is affine if it satisfies the property
$$f[ax + (1-a)y] = af(x) + (1-a)f(y).$$
Once again let's try $a=-1$, $x=1$, $y=0$:
$$\begin{align*}
|ax+(1-a)y| = |-1| &= 1\\
af(x)+(1-a)f(y) = -1\ |1| + 2\cdot 0 &= -1,\end{align*}$$
so $|x|$ isn't affine either.
A: Linear functions in analytic geometry are functions of the form $f(x)=a\cdot x+b$ for $a,b \in \mathbb{R}$.
Now try to write $\text{abs}(x)$ in such a form. 
Another way to see it: linear functions are "straight lines" in the coordinate system (excluding vertical lines), this clearly excludes having a "sharp edge" in the graph of the function like $\text{abs}(x)$ has it for $x=0$.
In linear algebra (and this is the more common definition) linear functions denote ones of the form $f(x)=a\cdot x$ which is equivalent to require $b=0$ in the above definition. As $\text{abs}(x)$ is not linear with the first, weaker definition it cannot be linear either with this definition.
A: I would simply define a linear function as always having the same slope (and, more technically, it must be continuous).
Clearly, the absolute value function has a negative slope for values < 0 and positive slope for values > 0. So it's not linear.
A: Think of the definition of absolute value.  It is a piecewise-defined function.
$|x| = x$, if $x\ge 0$, and $-x$ if $x<0$.  In other words, the graph of $y=|x|$ is formed by two pieces of two lines.
For the part of the domain where $x$-values are less than zero, the graph corresponds to the graph of $y=-x$.  For parts of the domain where $x$-values are greater than or equal to zero, the graph corresponds to the graph of $y=x$.
While the absolute value function does not satisfy the above definitions for linear functions, it is actually "parts" of two linear functions.
