What is are the solutions of #x#? #|x^2 + 3x| + x^2 - 2,>= 0#

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Answer 1 · 2017-12-21T09:30:01

#x in(-oo,-2/3]#

#|\ x^2 +3x|+x^2-2 >= 0#

#|\ x^2 +3x|>= -x^2+2 => ({(color(red)(x^2 +3x >= -x^2+2)) ,(color(blue)(x^2 +3x <= +x^2-2) )) #

#color(red)(x^2 +3x >= -x^2+2)#

#x^2 +3x+x^2-2 >= 0#

#2x^2 +3x-2 >= 0#

#Delta= 3^2-4.2.(-2)=25 #

#x_1= (-3-sqrt(25))/4=-2 #
#x_2= (-3+sqrt(25))/4=1/2 #
enter image source here

#color(red)(x in(-oo,-2]uu[1/2,+oo))#

#color(blue)(x^2 +3x <= +x^2-2)#
#3x <= -2 =>x <= -2/3 #
#color(blue)(x in(-oo,-2/3])#

#((color(red)(x in(-oo,-2]uu[1/2,+oo))),(color(blue)(x in(-oo,-2/3])))nnn=> x in(-oo,-2/3]#