How do you solve the inequality #abs(2x) + 6 < 22#?

1 Answer
smendyka
Jun 16, 2018

See a solution process below:

Explanation:

First, subtract #color(red)(6)# from each side of the inequality to isolate the absolution value function while keeping the inequality balanced:

#abs(2x) + 6 - color(red)(6) < 22 - color(red)(6)#

#abs(2x) + 0 < 16#

#abs(2x) < 16#

The absolute value function takes any term and transforms it to its non-negative form. Therefore, we must solve the term within the absolute value function for both its negative and positive equivalent.

#-16 < 2x < 16#

Divide each segment of the system of inequalities by #color(red)(2)# to solve for #x# while keeping the system balanced:

#-16/color(red)(2) < (2x)/color(red)(2) < 16/color(red)(2)#

#-8 < (color(red)(cancel(color(black)(2)))x)/cancel(color(red)(2)) < 8#

#-8 < x < 8#

Or

#x > -8#; #x < 8#

Or, in interval notation:

#(-8, 8)#

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