How do you solve the inequality #abs(3x-5)<=4# and write your answer in interval notation?

1 Answer
Sep 26, 2017

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.

#-4 <= 3x - 5 <= 4#

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

#-4 + color(red)(5) <= 3x - 5 + color(red)(5) <= 4 + color(red)(5)#

#1 <= 3x - 0 <= 9#

#1 <= 3x <= 9#

Now, divide each segment by #color(red)(3)# to solve for #x# while keeping the system balanced:

#1/color(red)(3) <= (3x)/color(red)(3) <= 9/color(red)(3)#

#1/3 <= (color(red)(cancel(color(black)(3)))x)/cancel(color(red)(3)) <= 3#

#1/3 <= x <= 3#

Or

#x >= 1/3# and #x <= 3#

Or, in interval notation:

#[1/3, 3]#