There is a multiple choices which says what is the order of $\{x\in\mathbb {Z}, |x|+|3x-1|<5\}$?
a. 1
b. 3
c. 2
d. empty
I know that by considering certain cases, for example when $x<0$ or $x\le 1/3$, we can find the answer which is $2$. I am just curious if there is a faster way than mine to find the solutions? Indeed, it is a multile choices and so we haven't enough time. Thank you!
Handle it case by case: $$ x\ge0\text{ and }x\ge\frac13\text{ and }4x-1\lt5\implies x\in\left[\frac13,\frac32\right) $$ $$ x\ge0\text{ and }x\lt\frac13\text{ and }1-2x\lt5\implies x\in\left[0,\frac13\right) $$ $$ x\lt0\text{ and }x\ge\frac13\text{ and }2x-1\lt5\implies x\in\varnothing $$ $$ x\lt0\text{ and }x\lt\frac13\text{ and }1-4x\lt5\implies x\in\left(-1,0\right) $$ Put it together: $$ x\in\left(-1,\frac32\right)\implies x\in\{0,1\} $$