How do I find the solutions to #1 <= abs(x-2) <= 4# algebraically?

1 Answer
Douglas K.
Oct 3, 2017

Use the piecewise definition for the absolute value function, #|f(x)| = {(f(x); f(x)>=0),(-f(x); f(x) < 0):}#, to separate the given inequality into two inequalities:

#1 <= -(x-2) <= 4# and #1 <= x-2 <= 4#

Multiply everything in the first inequality by -1:

#-1 >= x-2 >= -4#

Add two to everything:

#1 >= x >= -2#

We can flip this around:

#-2 <= x <= 1#

Add 2 to everything in the second inequality:

#3 <= x <= 7#

Combine the two inequalities:

#-2 <= x <= 1# and #3 <= x <= 7#

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