How do you solve and write the following in interval notation: #| 2x - 3 |<4#?

1 Answer
Jun 24, 2016

Answer:

The absolute value of a number depends on whether the number is positive, zero or negative. Then you have to take the given inequality into account.

Explanation:

#| a | = a#, if #a > 0#
#| a | = -a#, if #a < 0#
#| 0 | = 0#

There are then 3 cases to have a look at:
#2x - 3 > 0#, thus #2x > 3#, then #x>3/2#

In this case #|2x-3| = 2x-3#
and the inequality is, #2x - 3 < 4#, so
#2x < 4 + 3#, or #2x < 7#, that is
#x < 7/2#.
Thus, at the same time , #3/2 < x# and #x<7/2#, thus the elements #x# between #3/2# and #7/2# satisfy the inequality.

Similarly, if #2x - 3 < 0#, thus #2x < 3#, then #x<3/2#
In this case #|2x-3| = -2x+3#
and the inequality is, #-2x + 3 < 4#, so
#-2x < 4 - 3#, or #-2x < 1#, that is
#-1< 2x# and thus #-1/2 < x#
Then, at the same time , #-1/2 < x# and #x<3/2#, thus the elements #x# between #-1/2# and #3/2# satisfy the inequality.

Finally, when #2x-3=0#, #x=3/2#, and plugging the value #3/2# into the inequality we get #0 < 4#, which is true, and then #x=3/2# also satisfies the inequality.

Summarising, ALL the elements #x# between #-1/2# and #7/2# satisfy the inequality; that is the open interval (#-1/2#, #7/2#)