How do you solve #4< -z -4 < 11#?

1 Answer
Zor Shekhtman
Aug 19, 2015

#-15 < z < -8#

Explanation:

There are two inequalities here.
Firstly, let's solve them. Secondly, we will combine them into a resulting inequality for #z#.

  1. #4 < -z - 4#
    To solve this inequality for #z#, add #z# to both sides of equation and then subtract #4# from both sides.
    The first transformation will bring positive #z# to the left side instead of negative in the right, getting
    #z+4 < z-z-4#
    #z+4 < -4#
    The second transformation will get rid of #4# on the left:
    #z+4-4 < -4-4#
    #z < -8#

  2. #-z-4 < 11#
    To solve this inequality for #z#, add #z# to both sides of equation and then subtract #11# from both sides.
    The first transformation will bring positive #z# to the right side instead of negative in the left, getting
    #z-z-4 < z+11#
    #-4 < z+11#
    The second transformation will get rid of #11# on the right:
    #-4-11 < z+11-11#
    #-15 < z# or, equivalently, #z > -15#

So, we have two conditions on #z#:
#z < -8# and #z > -15#
We can combine them into one:
#-15 < z < -8#

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