The expression #color(blue)(-(x+6)(x+2)(x-2))#
is obviously equal to #color(blue)(0)# at #x in {color(magenta)(-6),color(magenta)(-2),color(magenta)(+2)}#
This gives us the critical points where the value of the expression might change between positive and negative (where the inequality #< 0# might apply).
We can select any other values below this set, between values in this set, and above this set to determine if the expression is #> 0# between these critical values.
For example:
#{:
("critical value at "x=,,color(magenta)(-6),,color(magenta)(-2),,color(magenta)(+2),),
("sample value for "x,=color(green)(-7),,color(green)(-4),,color(green)0,,color(green)(+4))
:}#
Evaluating at these sample values:
#{:
("when "x=," | ",-(x+6)(x+2)(x-2)=," | ","is this "> 0?),
(color(green)(-7)," | ",+45," | ",color(red)("Yes")),
(color(green)(-4)," | ",-24," | ","No"),
(color(white)(.)color(green)0," | ",+24," | ",color(red)("Yes")),
(color(green)(+4)," | ",-45," | ","No")
:}#
This gives us that
for #color(red)(x < -6)#
#color(white)("XXX")color(red)(-(x+6)(x+2)(x-2) > 0)#
for #-6 < x < -2#
#color(white)("XXX")-(x+6)(x+2)(x-2) cancel(>) 0#
for #color(red)(-2 < x < +2)#
#color(white)("XXX")color(red)(-(x+6)(x+2)(x-2) > 0)#
for #x > +2#
#color(white)("XXX")-(x+6)(x+2)(x-2) cancel(>) 0#
[note that at #x=-6 or -2 or +2# the expression is #=0# and
therefore it can not be #> 0# for these values.
