First, add #color(red)(2)# to each side of the inequality to isolate the absolute value term while keeping the inequality balanced:
#4abs(x + 1) - 2 + color(red)(2) < 10 + color(red)(2)#
#4abs(x + 1) - 0 < 12#
#4abs(x + 1) < 12#
Now, divide each side of the inequality by #color(red)(4)# to isolate the absolute value function while keeping the inequality balanced:
#(4abs(x + 1))/color(red)(4) < 12/color(red)(4)#
#(color(red)(cancel(color(black)(4)))abs(x + 1))/cancel(color(red)(4)) < 3#
#abs(x + 1) < 3#
The absolute value function takes any term and transforms it to its non-negative form. Therefore, we must solve the term within the absolute value function for both its negative and positive equivalent.
#-3 < x + 1 < 3#
Now, subtract #color(red)(1)# from each segment of the system of inequalities to solve for #x# while keeping the system balanced:
#-3 - color(red)(1) < x + 1 - color(red)(1) < 3 - color(red)(1)#
#-4 < x + 0 < 2#
#-4 < x < 2#
Or
#x > -4# and #x < 2#
Or, in interval notation:
#(-4, 2)#
