Question

The minimum value of the expression $y=|x+3|+|x+1|+|x-5|$ is

Answer 1

$y=|x+3|+|x+1|+|x-5|$
Here the turning points are $-3,-1,5$

$|x+3|+|x+1|+|x-5|=\{\begin{array}{cc}-(x+3)-(x+1)-(x-5),& x<-3\\ (x+3)-(x+1)-(x-5),& -3\le x<-1\\ (x+3)+(x+1)-(x-5),& -1\le x<5\\ (x+3)+(x+1)+(x-5),& x\ge 5\end{array}$

$\Rightarrow |x+3|+|x+1|+|x-5|=\{\begin{array}{cc}1-3x,& x<-3\\ 7-x,& -3\le x<-1\\ 9+x,& -1\le x<5\\ 3x-1,& x\ge 5\end{array}$


Hence, the minimum value of the given expression is $8$.