What is the solution set for #absx - 1 < 4#?

1 Answer
Stefan V.
Aug 14, 2015

#-5 < x < 5#

Explanation:

To solve this absolute value inequality, first isolate the modulus on one side by adding #1# to both sides of the inequality

#|x| - color(red)(cancel(color(black)(1))) + color(red)(cancel(color(black)(1))) < 4 + 1#

#|x| < 5#

Now, depending on the possible sign of #x#, you have two possiblities to account for

  • #x>0 implies |x| = x#

This means that the inequality becomes

#x < 5#

  • #x<0 implies |x| = -x#

This time, you have

#-x < 5 implies x> -5#

These two conditions will determine the solution set for the absolute value inequality. Since the inequality holds true for #x> -5#, any value of #x# that's smaller than that will be excluded.

LIkewise, since #x<5#, any value of #x# greater than #5# will also be excluded. This means that the solution set to this inequality will be #-5 < x < 5#, or #x in (-5, 5)#.

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