Question

Solve the equation
$|x-1|+|7-x|+2|x-2|=4$. The number of solutions is

Answer 1

Here critical points are $1,2,7$ using the method of intervals,
We find intervals when the expressions $x-1,7-x$ and $x-2$ are of constant signs $x<1,17$
The given equation is equivalent to the collection of four systems,
$\{\begin{array}{c}-(x-1)+(7-x)-2(x-2)=4,x<1\\ (x-1)+(7-x)-2(x-2)=4,1\le x\le 2\\ (x-1)+(7-x)+2(x-2)=4,2\le x\le 7\\ (x-1)-(7-x)+2(x-2)=4,x\ge 7\end{array}$

$\{\begin{array}{c}x=2,x<1\\ x=3,1\le x\le 2\\ x=1,2\le x\le 7\\ x=4,x\ge 7\end{array}$

From the collection of four systems, the given equation has no solution.