SOLUTION: Solve the following inequality {{{ sqrt(7-x) }}} > {{{ x-1 }}}

1.  The domain of this inequality is the set  7-x >= 0,  or   x <= 7.



2.  All the set  x-1 < 0,  or  x < 1,  is the solution, BECAUSE the right side is negative there, 

                                                         while the left side is not negative.


    Therefore, we need analyze for the solution only the remaining part  1 <= x <= 7.

    Let's call this set { 1 <= x <= 7 } as A17.



3.  Next, analyzing for the set A17, square both sides of the original inequality.  You will get


        7 - x > x^2 - 2x + 1

        x^2 - x - 6 < 0

        (x-3)*(x+2) < 0


    The solution set to the last inequality  is   -2 < x < 3.      

    Taking the intersection with the set A17, we have the solution set  { 1 <= x <3 }.



4.  Thus we have two parts of the entire solution set  (a)  { x < 1 }  from n.2,  and  (b) { 1 <= x < 3 } from n.3.


     The final solution set is the union  sets (a) and (b), i.e.  { x < 1 } U { 1 <= x < 3},  which is,  OBVIOUSLY,  (-oo < x < 3 }.



ANSWER.  The solution to the original inequality is the set  { -oo < x < 3 },  or  (-oo,3).



                          Visual CHECK



    


    Plot  y =  (red line)  and  y = x-1  (green line)



The solution set is the set of all points of the x-axis, where red line is above the green line.    

The post-solution note