Because the absolute value function takes and negative or positive term and transforms it into the positive form of the term we must solve the term inside the absolute value function for both the negative and positive equivalent:
#-10 > 1 - 2x > 10#
Now, subtract #color(red)(1)# from each segment of the system of inequalities to isolate the #x# term while keeping the system balanced:
#-10 - color(red)(1) > -color(red)(1) + 1 - 2x > 10 - 1#
#-11 > 0 - 2x > 9#
#-11 > -2x > 9#
Now, divide each segment by #color(blue)(-2)# to solve for #x# while keeping the system balanced. However, because we are multiplying or dividing an inequality by a negative term we must reverse the inequality operators:
#(-11)/color(blue)(-2) color(red)(<) (-2x)/color(blue)(-2) color(red)(<) 9/color(blue)(-2)#
#11/2 color(red)(<) (color(blue)(cancel(color(black)(-2)))x)/cancel(color(blue)(-2)) color(red)(<) -9/2#
#11/2 color(red)(<) x color(red)(<) -9/2#
Or
#x > 11/2# and #x < -9/2#
Or, in interval notation:
#(-oo, -9/2)# and #(11/2, +oo)#
