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

1 Answer
Jan 23, 2018

Answer:

See a solution process below: #(-2; 20/7)#

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.

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.

enter image source here