First, add #color(red)(3)# to each side of the inequality to isolate the absolute value term while keeping the equation balanced:
#2abs(4b) - 3 + color(red)(3) > 77 + color(red)(3)#
#2abs(4b) - 0 > 80#
#2abs(4b) > 80#
Next, divide each side of the inequality by #color(red)(2)# to isolate the absolute value function while keeping the equation balanced:
#(2abs(4b))/color(red)(2) > 80/color(red)(2)#
#(color(red)(cancel(color(black)(2)))abs(4b))/cancel(color(red)(2)) > 40#
#abs(4b) > 40#
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.
#-40 > 4b > 40#
Divide each segment of the system of inequalities by #color(red)(4)# to solve for #b# while keeping the system balanced:
#-40/color(red)(4) > (4b)/color(red)(4) > 40/color(red)(4)#
#-10 > (color(red)(cancel(color(black)(4)))b)/cancel(color(red)(4)) > 10#
#-10 > b > 10#
Or
#b < -10#; #b > 10#
Or, in interval notation:
#(-oo, -10); (10, +oo)#
