I need to find the supremum and infimum of the following set: $$\Big\{ y \in \mathbb{R} \Big| y = x + |x - 1| \text{ for some } x \in \mathbb{R}\Big\}$$
I'm unable to use limits or differentiation for questions like this one and I'm unsure of what to do in order to solve it. If anyone is able to help me I would greatly appreciate it, thanks!
By inspection of the equation I know when $x$ is really large and positive that there's not going to be a supremum, and when $x$ is large and negative it's going to tend towards 0. But I'm not really sure how to show this without taking limits and I'm a bit unsure how to proceed.
Consider two separate cases. The first is $x \ge 1$ and the second is $x <1$.
If $x \ge 1$, then $$y=x + |x-1| = x+x-1 = 2x-1$$ Clearly this has no upper bound, while the lower bound is $y=1$ (when $x=1$).
If $x<1$, then $$y=x + |x-1| = x-x+1 = 1$$
In other words your set is $$\left\{ y \in \Bbb R | y \ge 1\right\}$$ which is the interval $[1, + \infty )$.