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.
#-4 >= -k - 7 >= 4#
First, add #color(red)(7)# to each segment of the system of inequalities to isolate the #k# term while keeping the system balanced:
#-4 + color(red)(7) >= -k - 7 + color(red)(7) >= 4 + color(red)(7)#
#3 >= -k - 0 >= 11#
#3 >= -k >= 11#
Now, multiply each segment by #color(blue)(-1)# to solve for #k# while keeping the system balanced. However, because we are multiplying or dividing inequalities by a negative number we must reverse the inequality operators:
#color(blue)(-1) xx 3 color(red)(<=) color(blue)(-1) xx -k color(red)(<=) color(blue)(-1) xx 11#
#-3 color(red)(<=) k color(red)(<=) -11#
Or
#k <= -11#; #k >= -3#
Or, in interval notation:
#(-oo, -11]#; #[-3, +oo)#
