How do you solve the inequality #abs(2x-3)>=5#?

1 Answer
smendyka
Jul 25, 2017

See a solution process below:

Explanation:

The absolute value function takes any negative or positive 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.

#-5 >= 2x - 3 >= 5#

First, add #color(red)(3)# to each segment of the system of inequalities to isolate the #x# term while keeping the system balanced:

#-5 + color(red)(3) >= 2x - 3 + color(red)(3) >= 5 + color(red)(3)#

#-2 >= 2x - 0 >= 8#

#-2 >= 2x >= 8#

Now, divide each segment by #color(red)(2)# to solve for #x# while keeping the system balanced:

#-2/color(red)(2) >= (2x)/color(red)(2) >= 8/color(red)(2)#

#-1 >= (color(red)(cancel(color(black)(2)))x)/cancel(color(red)(2)) >= 4#

#-1 >= x >= 4#

Or

#x <= -1#; #x >= 4#

Or, in interval notation:

#(-oo, -1]; [4, +oo)#

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