Are there any $x,y,z$ in $\mathbb{R}$ for which the following equations hold?

Question

Are there any $x,y,z \in \mathbb{R}$ for which the following equations hold? $$|x+1| \leq 2\\ |y+1| \leq 3\\ |y-z| \leq 1$$

With the given we know that $x$ is between $[-3,1]$, $y$ is between $[-4,2]$, and $(y-z)$ is between $[-1,1]$.

Using a triangle inequality:

$$|(x+1)+(-y-1)|= |x-y| \leq |x+1| + |y+1|$$

there i can see that |x-y| can be maximum 7; and on the right side |x+1| can me maximum 2; and |y+1| maximum 3; which is a contradiction since 7 is never less than 5.

maybe this is proven but am not sure The anwser to this question is no.

any feedback is welcome Thanks.

Answer 1

For every $$x \in [-3,1]$$ and $$y \in [-4,2]$$ it is possible to select some $z$: $$z \in [y-1,y+1]$$