Question 40809
f(x) = |x-4| - |x+5|


When x<-5, |x-4| = -(x-4) = 4-x and |x+5| = -(x+5) = -x-5
So f(x) = |x-4| - |x+5| = 4-x+x+5 = 9 


When -5<x<4, |x-4| = -(x-4) = 4-x and |x+5| = x+5
So f(x) = |x-4| - |x+5| = 4-x-x-5 = -2x-1


When 4<x, |x-4| = x-4 and |x+5| = x+5
So f(x) = |x-4| - |x+5| = x-4-x-5 = -9


When x = -5, f(x) = |-5-4| = 9
When x = 4, f(x) = - |4+5| = -9


So f(x) = 9 when x <= -5
and f(x) = -9 when x >= 4


Thus the interval in which f(x) is constant is x <= -5 and x >= 4.