Given: #-5|-2x -3| + 2 = -33#
Subtract 2 from both sides:
#-5|-2x + -3| = -35#
Divide both sides by -5:
#|-2x + -3| = 7" [1]"#
Using the definition of the absolute value function,
#|A| = {(A; A>=0),(-A;A < 0):}#
We can write the following:
#|-2x -3| = {(-2x-3; -2x-3>= 0),(2x+3;-2x-3<0):}#
Simplify the inequalities:
#|-2x -3| = {(-2x-3; -2x>= 3),(2x+3;-2x<3):}#
#|-2x -3| = {(-2x-3; x<= -3/2),(2x+3;x> -3/2):}#
Substitute both into equation [1]:
#-2x -3 = 7; x <= -3/2# and #2x + 3 = 7; x > -3/2#
#-2x = 10; x <= -3/2# and #2x = 4; x > -3/2#
#x = -5; x <= -3/2# and #x = 2; x > -3/2#
Check in the original equation:
#-5|-2(-5) -3| + 2 = -33# and #-5|-2(2) -3| + 2 = -33#
#-5|7| + 2 = -33# and #-5|-7| + 2 = -33#
#-33 = -33# and #-33 = -33#
Both values check.
#x = -5# and #x = 2#
