Solve for $x$ :$|[x]-2x|=4$

Question

Solve for $x$ $$|[x]-2x|=4$$ where $[.]$ denotes greatest integer function. How to solve it graphically? can any one guide me step by step to get the solution ?

Answer 1

Without Using Graph.

Given $\left|\lfloor x \rfloor -2x\right| = 4\Rightarrow \left|-\lfloor x \rfloor -2\left\{x\right\}\right|=4\Rightarrow \left|\lfloor x \rfloor +2\{x\}\right|=4$

Now Here Right hand Side is Integer. So Left hand Side must be integer.

So Using $0 \leq \{x\}<1\Rightarrow 0\leq 2\{x\}<2.$

So we get $2\{x\} =0\Rightarrow \{x\} =0$ or $\displaystyle 2\{x\} = 1\Rightarrow \{x\} = \frac{1}{2}$

$\bullet\; $ If $\{x\}=0\;,$ Then equation convert into $|\lfloor x \rfloor | = 4\Rightarrow |x|=4\Rightarrow x=\pm 4$

$\bullet\; $ If $\displaystyle \{x\}=\frac{1}{2}\;,$ Then equation convert into $|\lfloor x \rfloor +1| = 4\Rightarrow \lfloor x\rfloor =\pm 4-1\Rightarrow \lfloor x\rfloor =3,-5$

So we get $\displaystyle x = \lfloor x \rfloor +\{x\} = 3+\frac{1}{2}\;\;,-5+\frac{1}{2}=3.5\;\;,-4.5$

So we get $\displaystyle x = \left\{\pm 4\;, 3.5\;,-4.5\right\}$

Answer 2

Solving graphically would involve sketching the graph o fthe function. But sketches are no solution after all.

As $[x]$ and $4$ are integers, we conclude that $2x$ is an integer. Also $x-1<[x]\le x$ make $-x-1<[x]-2x\le -x$, so that either $-4\le -x<-3$ or $4\le -x<5$. These conditions leave us with $x\in\{-4, -3\tfrac12,4,4\tfrac12\}$ to verify.