How do you solve and graph #abs(5-x)>=3#?

1 Answer
Jul 27, 2017

Answer:

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.

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

First, subtract #color(red)(5)# from each segment of the system of inequalities to isolate the #x# term while keeping the system balanced:

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

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

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

Now, multiply each segment by #color(blue)(-1)# to solve for #x# while keeping the system balanced. However, because we are multiplying or dividing inequalities by a negative number we must reverse the inequality operators:

#color(blue)(-1) xx -8 color(red)(<=) color(blue)(-1) xx -x color(red)(<=) color(blue)(-1) xx -2#

#8 color(red)(<=) x color(red)(<=) 2#

Or

#x <= 2#; #x >= 8#

Or, in interval notation:

#(-oo, 2]#; #[8, +oo)#