How do you solve the inequality #|1-x/3|>2/3#?

1 Answer
sente
Dec 1, 2015

Consider each case for the inequality being true to find the solution set #(-oo,1)uu(5,oo)#

Explanation:

We can solve this if we recognize what the absolute value symbol is actually saying. It is equivalent to saying

#1-x/3 > 2/3#
or
#1-x/3 < -2/3#

Then all we need to do is find what conditions cause each to be true.

For the first:
#1-x/3 > 2/3#
#=> 1-2/3 > x/3#
#=> 1/3 > x/3#
#=> 1 > x#

For the second:
#1-x/3 < -2/3#
#=> 1+2/3 < x/3#
#=> 5/3 < x/3#
#=> 5 < x#

So one of the conditions, and thus the original inequality, will be true so long as either #x < 1# or #x > 5#. Therefore we have the solution set
#(-oo,1)uu(5,oo)#

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