Question about X+Y interval

Question

I just have a question for

Q: Let X be the number on a die roll between 1 and 6. Let Y be a random number which is uniformly distributed on [0,1] independent of X Let Z = 10 X + 10 Y

for this question how come the range of X + Y is the interval [1,7] ?

Answer 1

The absolute minimum of $x + y$ is when $x = 1$ and $y = 0$ which gives $x + y = 1$. The absolute maximum of $x + y$ is when $x = 6$ and $y = 1$ which gives $x + y = 7$. Since $y$ can be any real number between $0$ and $1$, $x + y$ can be any real number in the interval $[1, 7]$. $Z$ does not matter in this problem.

Answer 2

$$ 1=1+0=\min X +\min Y\le X+Y\le\max X+\max Y=6+1=7. $$