How do you graph and solve # | 3 - 7 x |>17#?

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.

We can rewrite and solve this inequality as:

#-17 > 3 - 7x > 17#

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

#-17 - color(red)(3) > 3 - color(red)(3) - 7x > 17 - color(red)(3)#

#-20 > 0 - 7x > 14#

#-20 > -7x > 14#

Now, divide each segment by #color(blue)(-7)# to solve for #x# while keeping the system balanced. Because we are multiplying or dividing an inequality by a negative number we must reverse the inequality operators.

#(-20)/color(blue)(-7) color(red)(<) (-7x)/color(blue)(-7) color(red)(<) 14/color(blue)(-7)#

#20/7 color(red)(<) (color(blue)(cancel(color(black)(-7)))x)/cancel(color(blue)(-7)) color(red)(<) -2#

#20/7 color(red)(<) x color(red)(<) -2#

Or.

#x < -2#; #x > 20/7#

Or, in interval notation:

#(-oo, -2)#; #(20/7, +oo)#

To graph this we will draw a vertical lines at #-2# and #20/7# 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 outside the two lines to show the interval where the inequality is true.