integer roots

[kinu]

find Integer value of \(\displaystyle a\) for which the equation \(\displaystyle |2x+1|+|x-2|=a\) have only integer roots.

 

[daon]

If I am interpreting this correctly, I would start with partitioning the real line by the zeros of the function stuck into the absolute values: -1/2 and 2

x <= -1/2

You get: -(2x+1)-(x-2)=a

-1/2 < x <= 2

You get: (2x+1)-(x-2)=a

x > 2

You get: (2x+1)+(x-2)=a

In the first case: -3x+1=a gives the only possible integer values a=1-3k for which x=k will be a root.

Try the others.