First, group and combine like terms in the middle segment of the system of inequalities:
#-14 <= 7k + 1 - 10k <= 7#
#-14 <= 7k - 10k + 1 <= 7#
#-14 <= (7 - 10)k + 1 <= 7#
#-14 <= -3k + 1 <= 7#
Next, subtract #color(red)(1)# from each segment to isolate the #k# term while keeping the system balanced:
#-14 - color(red)(1) <= -3k + 1 - color(red)(1) <= 7 - color(red)(1)#
#-15 <= -3k + 0 <= 6#
#-15 <= -3k <= 6#
Now, divide each segment by #color(blue)(-3)# 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:
#(-15)/color(blue)(-3) color(red)(>=) (-3k)/color(blue)(-3) color(red)(>=) 6/color(blue)(-3)#
#5 color(red)(>=) (color(red)(cancel(color(black)(-3)))k)/cancel(color(blue)(-3)) color(red)(>=) -2#
#5 color(red)(>=) k color(red)(>=) -2#
Or
#k > -2# and #k <= 5#
Or, in interval notation:
#[-2, 5]#
