Here we have to solve the inequality #(x+7)(2x+8)(3x-12)>=0# or #(x+7)xx2(x+4)xx3(x-4)>=0# or #6(x+7)(x+4)(x-4)>=0# or #(x+7)(x+4)(x-4)>=0#
From this we know that the product #(x+7)(x+4)(x-4)>=0# is positive or zero. It is apparent that sign of binomials #(x+7)#, #(x+4)# and #(x-4)# will change around the values #-7#. #-4# and #4# respectively.
In sign chart we divide the real number line in four parts, i.e. below #-7#, between #-7# and #-4#, between #-4# and #4# and above #4# and see how the sign of #(x+7)(x+4)(x-4)# changes.
Sign Chart
#color(white)(XXXXXXXXXXX)-7color(white)(XXXXX)-4color(white)(XXXXX)4#
#(x+7)color(white)(XXXX)-ive color(white)(XXXX)+ive color(white)(XXX)+ive color(white)(XXX)+ive#
#(x+4)color(white)(XXXX)-ive color(white)(XXXX)-ive color(white)(XXX)+ive color(white)(XXX)+ive#
#(x-4)color(white)(XXXX)-ive color(white)(XXXX)-ive color(white)(XXX)-ive color(white)(XXX)+ive#
#(x+7)(x+4)(x-4)#
#color(white)(XXXXXXXX)-ive color(white)(XXXX)+ive color(white)(XX)-ive color(white)(XXX)+ive#
It is observed that #(x+7)(x+4)(x-4)>=0# when either #-7 <= y <= -4# or #y >= 4#, which is the solution for the inequality.
