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.
#-9 < (3x - 6)/4 < 9#
First, multiply each segment of the system of inequalities by #color(red)(4)# to eliminate the fraction while keeping the system balanced:
#color(red)(4) xx -9 < color(red)(4) xx (3x - 6)/4 < color(red)(4) xx 9#
#-36 < cancel(color(red)(4)) xx (3x - 6)/color(red)(cancel(color(black)(4))) < 36#
#-36 < 3x - 6 < 36#
Next, add #color(red)(6)# to each segment to isolate the #x# term while keeping the system balanced:
#-36 + color(red)(6) < 3x - 6 + color(red)(6) < 36 + color(red)(6)#
#-30 < 3x - 0 < 42#
#-30 < 3x < 42#
Now, divide each segment by #color(red)(3)# to solve for #x# while keeping the system balanced:
#-30/color(red)(3) < (3x)/color(red)(3) < 42/color(red)(3)#
#-10 < (color(red)(cancel(color(black)(3)))x)/cancel(color(red)(3)) < 14#
#-10 < x < 14#
Or
#x > -10#; #x < 14#
Or, in interval notation:
#(-10, 14)#
