First, add #color(red)(5)# to each side of the inequality to isolate the absolute value function while keeping the inequality balanced:
#abs(2x + 1) - 5 + color(red)(5) < 0 + color(red)(5)#
#abs(2x + 1) - 0 < 5#
#abs(2x + 1) < 5#
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.
#-5 < 2x + 1 < 5#
Next, subtract #color(red)(1)# from each segment of the system of inequalities to isolate the #x# term while keeping the system balanced:
#-5 - color(red)(1) < 2x + 1 - color(red)(1) < 5 - color(red)(1)#
#-6 < 2x + 0 < 4#
#-6 < 2x < 4#
Now, divide each segment of the system by #color(red)(2)# to solve for #x# while keeping the system balanced:
#-6/color(red)(2) < (2x)/color(red)(2) < 4/color(red)(2)#
#-3 < (color(red)(cancel(color(black)(2)))x)/cancel(color(red)(2)) < 2#
#-3 < x < 2#
Or
#x > -3# and #x < 2#
Or, in interval notation:
#(-3, 2)#
