Inequality...Assume b < c

Solve the inequality and write answer in interval notation.

|x - a| + b < c

|x-a|+b
|x-a|
-(c-b)<(x-a)
b-c<(x-a)
a+b-c
(a+b-c,a+c-b)

|x - a| + b < c
|x - a| < c - b
(x - a)^2 < (c - b)^2
x^2 - 2ax + a^2 - (c - b)^2 < 0
consider x^2 - 2ax + a^2 - (c - b)^2 = 0
x = a +/- sqrt(4a^2 - 4(a^2 - (c - b)^2))/2
x = a +/- (c - b)

x = a + c - b is a solution and x = a - c + b is a solution

therefore a - c + b < x < a + c - b

|x - a| + b < c
|x - a| < c - b [Since b < c, c - b > 0]
-(c - b) < x - a < c - b
-c + b < x - a < c - b
a + b - c < x < a - b + c
x ∊ (a + b - c, a - b + c)

Solve for what?