As we know the following property about absolute value:
#color(red)(absa=-a if a<0)#
#color(blue)(absa=a if a>0)#
we will use this property about inequalities:
#color(purple)(ax< b) #
#color(purple)(x< b/a if a> 0) #
#color(brown)(x> b/a if a< 0) #
In the given exercise because #x# is unknown so we have to solve the inequality based on the above conditions
#color(red)(abs(2x+9)=-(2x+9)= if 2x+9<0)#
#abs(2x+9)=-2x-9 if 2x<-9#
#color(red)(abs(2x+9)=-2x-9 if x<-9/2)#
#color(blue)(abs(2x+9)=2x+9 if 2x+9>0)#
#color(blue)(abs(2x+9)=2x+9 if x> -9/2)#
So,
#6abs(2x+9)-14<=16#
#rArr6color(blue)((2x+9))-14<=16 color(blue)(if x> -9/2)#
#rArr12x+54-14<=16 color(blue)(if x> -9/2)#
#rArr12x<=16 -54+14color(blue)(if x> -9/2)#
#rArr12x<=-24color(blue)(if x> -9/2)#
#rArrx<=-24/12#
#rArrx<=-2color(blue)(if x> -9/2)#
#6abs(2x+9)-14<=16#
#rArr6color(red)((-2x-9))-14<=16 color(red)(if x<-9/2)#
#rArr-12x-54-14<=16 color(red)(if x< -9/2)#
#rArr-12x<=16 +54+14color(red)(if x< -9/2)#
#rArr-12x<=84color(red)(if x< -9/2)#
#rArrxcolor(brown)>=84/(-12)color(red)(if x< -9/2)#
#rArrx>=-7color(red)(if x< -9/2)#
Therefore, the solution is that
#x<=-2color(blue)(if x> -9/2)#
#x>=-7color(red)(if x< -9/2)#
