How do you solve #abs(x - 7)<10#?

1 Answer
Alan P.
Apr 7, 2015

We have two situations
(1) #x-7 < 0#
and
(2) #x-7 >= 0#

Case (1)
If #x-7 < 0# which implies #x < 7#
then
#abs(x-7) < 10#
is equivalent to
#7-x < 10#
#-x < 3#
#x > -3#
so the inequality holds if #-3 < x < 7#

Case (2)
If #x-7 >= 0# which implies #x >= 7#
#abs(x-7) < 10#
is equivalent to
#x-7 <10#
#x<17#
so the inequality holds if #7 <= x < 17#

Solution
The solution to the given inequality includes all values in the 2 ranges
#-3 < x < 7#
and
#7 <= x < 17#
which can be combined into a single range
#-3 < x < 17#

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