First, add #color(red)(3)# to each side of the inequality to isolate the absolute value function while keeping the inequality balanced:
#abs(2x + 1) - 3 + color(red)(3) > 6 + color(red)(3)#
#abs(2x + 1) - 0 > 9#
#abs(2x + 1) > 9#
The absolute value function takes any number and transforms it to its non-negative form. Therefore we need to solve the term within the absolute value function for both the negative and positive value it is equated to.
#-9 > 2x + 1 > 9#
Next, subtract #color(red)(1)# from each segment of the system of inequalities to isolate the #x# term while keeping the system balanced:
#-9 - color(red)(1) > 2x + 1 - color(red)(1) > 9 - color(red)(1)#
#-10 > 2x + 0 > 8#
#-10 > 2x > 8#
Now, divide each segment by ## while keeping the system balanced:color(red)(2)
#-10/color(red)(2) > (2x)/color(red)(2) > 8/color(red)(2)#
#-5 > (color(red)(cancel(color(black)(2)))x)/cancel(color(red)(2)) > 4#
#-5 > x > 4#
Or
#x < -5# and #x > 4#
Or, in interval notation:
#(-oo, -5)# and #(4, +oo)#
