How do you graph and solve #|2x+3|<=15#?

1 Answer
smendyka
Jan 20, 2018

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.

#-15 <= 2x + 3 <= 15#

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

#-15 - color(red)(3) <= 2x + 3 - color(red)(3) <= 15 - color(red)(3)#

#-18 <= 2x + 0 <= 12#

#-18 <= 2x <= 12#

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

#-18/color(red)(2) <= (2x)/color(red)(2) <= 12/color(red)(2)#

#-9 <= (color(red)(cancel(color(black)(2)))x)/cancel(color(red)(2)) <= 6#

#-9 <= x <= 6#

Or

#x >= -9#; #x <= 6#

Or. in interval notation

#[-9, 6]#

To graph this we will draw vertical lines at #-9# and #6# on the horizontal axis.

The lines will be a solid line because the inequality operators contain "or equal to" clauses.

We will shade between the lines to show the interval:

enter image source here

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