First, expand the terms in parenthesis on each side of the equality by multiplying each term within the parenthesis by the term outside the parenthesis:
#5 - color(red)(2)(3 - x) <= color(blue)(2)(2x + 5) + 1#
#5 - (color(red)(2) xx 3) + (color(red)(2) xx x) <= (color(blue)(2) xx 2x) + (color(blue)(2) xx 5) + 1#
#5 - 6 + 2x <= 4x + 10 + 1#
#-1 + 2x <= 4x + 11#
Next, subtract #color(red)(2x)# and #color(blue)(11)# from each side of the inequality to isolate the #x# term while keeping the inequality balanced:
#-1 + 2x - color(red)(2x) - color(blue)(11) <= 4x + 11 - color(red)(2x) - color(blue)(11)#
#-1 - color(blue)(11) + 2x - color(red)(2x) <= 4x - color(red)(2x) + 11 - color(blue)(11)#
#-12 + 0 <= (4 - color(red)(2))x + 0#
#-12 <= 2x#
Now, divide each side of the inequality by #color(red)(2)# to solve for #x# while keeping the inequality balanced:
#-12/color(red)(2) <= (2x)/color(red)(2)#
#-6 <= (color(red)(cancel(color(black)(2)))x)/cancel(color(red)(2))#
#-6 <= x#
To state the solution in terms of #x# we can reverse or "flip" the entire inequality:
#x >= -2#
