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.
#-16 >= -4x + 7 >= 16#
First, subtract #color(red)(7)# from each segment of the system of inequalities to isolate the #x# term while keeping the system balanced:
#-16 - color(red)(7) >= -4x + 7 - color(red)(7) >= 16 - color(red)(7)#
#-23 >= -4x + 0 >= 9#
#-23 >= -4x >= 9#
Now, divide each segment by #color(blue)(-4)# to solve for #x# while keeping the system balanced. However, because we are dividing or multiplying inequalities by a negative number we must reverse the inequality operators:
#(-23)/color(blue)(-4) color(red)(<=) (-4x)/color(blue)(-4) color(red)(<=) 9/color(blue)(-4)#
#23/4 color(red)(<=) (color(blue)(cancel(color(black)(-4)))x)/cancel(color(blue)(-4)) color(red)(<=) -9/4#
#23/4 color(red)(<=) x color(red)(<=) -9/4#
Or
#x <= -9/4# and #x >= 23/4#
Or, in interval notation:
#(=oo, -9/4]# and #[23/4, +oo)#
