# Sum of two independent uniform variables explanation

I am currently studying for a final in my probability course and I am looking through my notes and there is a problem I don't exactly understand. The problem is shown below:

I understand the first step (comes form a formula), but I don't understand the progression from the second inequality onward. Any explanation would be great, thank you!

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$f_X(x)$ is zero if $x \notin [0,1]$ (and $f_X(x) = 1$ for $x \in [0,1]$). The same argument is used after: $f_Y(z -x)=0$ whenever $z - x \notin [0,1]$. Hope this helps. –  roger Dec 8 '13 at 17:36

I'm an electrical engineer and conceptually, commutation (which is what this is called) is analogous to sliding one waveform across the other. The "fixed" PDF is a square wave 1 unit high between 0 and 1. The "sliding" PDF is also 1 unit high and 1 unit wide and as it slides up from the left, the values are 0 until it just starts to overlap at $x=-1$, then it rises linearly so that when say 0.1 (at $x=-.9$) of the waves overlap the value is 0.1 and so on until they are fully superimposed at $x=0$. They them move apart linearly until they part company at $x=1$. This is why is is really easy to construct a triangle wave generator when you have 2 square wave generators.