If #abs(x-2)<6#, then what are the bounds of #x#? i.e. find #a# and #b# such that #a<x<b#.
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#abs(x-2) < 6# if and only if #-6 < x-2 < 6#
if and only if #-6+2 < x-2+2 < 6+2#
if and only if #-4 < x < 8#.
A graph often helps to visualise inequalities:
graph{(y-|x-2|)(y-6)=0 [-6.5, 13.5, -1.4, 8.6]}
#abs(x-2) < 6# is the region below the line #y=6# which you can see is the interval #-4 < x < 8#.
#abs (x-2) < 6" "<=>" ""-"6< x-2<6" "<=>" ""-"4< x<8#.
#a="-"4, b=8.#
When an absolute value inequality has an upper limit (like #abs (x-2) < 6# does), what it means is that the magnitude of #x-2# needs to be under 6; in other words, #x-2# can be positive or negative, but it must be within 6 units of 0, either way.
This means #x-2# has a lower limit of #-6#, but also an upper limit of #6#. We can then rewrite #abs(x-2)<6# as
#"-"6 < x-2 < 6#.
The final step is to isolate #x#. We do this by adding 2 to all "sides" of the inequality, to get
#"-"6+color(red)2 < x - 2 + color(red)2 < 6 + color(red)2 #
or
#"-"4< x < 8#.
Thus, the lower limit of #x# is #a="-"4#, and the upper limit of #x# is #b=8#.
