Walrasian demand with a twist of Leontief function

Why don't you just plug in some values for $x_1$ and $x_2$? Start with something like

$x_1=1, x_2=1$ and find utility $u(1,1)=1+5*1$.

Then increase $x_1$ or $x_2$ and let

$x_1=2, x_2=1$ and find utility $u(2,1)=1+5*2=11$, and

$x_1=1, x_2=2$ and find utility $u(1,2)=1+5*2=11$.

The utilities of both bundles are identical, and the goods seem to be substitutable. In fact for any $x_1<x_2$ you will find $u(x_1,x_2) = x_1 + 5 x_2$, and, similarly, for any $x_1>x_2$ you will find $u(x_1,x_2) = x_2 + 5 x_1$.

Now, just for now, assume both goods have the same price $p_1=p_2=1$ and you have a budget of $w=3$ such that you can buy both bundles giving utility 11 up there. You can see that it would be better to put all your money in one of the goods and get $u(3,0)=u(0,3) = 0 + 5*3=15$. Now think about different prices and a general income $w$. Does it again make sense to buy only one of the goods? Is it the cheaper one?