Solve : `x^(2)- |x+2|+x gt 0`

Explanation

We have `x^(2)- |x+2|+x gt 0`
or `" "x^(2)+x gt |x+2|" "` (i)
To solve the inequality, we draw the graph of `y=x^(2)+x and y= |x+2|` as shown in the following figure.

Solving `y= |x+2| and y=x^(2)+x` for their points of intersection A and B, we have `x+2= x^(2)+x rArr x^(2)=2 rArr x= pm sqrt2`
The solution set of (i) is the set of those values of x for which the graph of `y=x ^(2)+x` lies above the graph of `y=x +2`.
From the graph, the solution is `(-oo, -sqrt2) uu (sqrt2, oo)`.