How do you graph and solve #abs{4x -3} > 13#?

1 Answer
Jan 19, 2018

Answer:

See a solution process below:

Explanation:

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.

#-13 > 4x - 3 > 13#

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

#-13 + color(red)(3) > 4x - 3 + color(red)(3) > 13 + color(red)(3)#

#-10 > 4x - 0 > 16#

#-10 > 4x > 16#

Now, divide each segment by #color(red)(4)# to solve for #x# while keeping the system balanced:

#-10/color(red)(4) > (4x)/color(red)(4) > 16/color(red)(4)#

#-10/4 > (color(red)(cancel(color(black)(4)))x)/cancel(color(red)(4)) > 4#

#-5/2 > x > 4#

Or

#x < -5/2#; #x > 4#

Or, in interval notation:

#(-oo, -5/2)#; #(4, +oo)#

To graph this we will draw vertical lines at #-5/2# and #4# on the horizontal axis.

The lines will be a dashed lines because the inequality operators do not contain an "or equal to" clause.

We will shade to the left and right side of the lines respecitively:

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