How do you solve and graph #abs(3d-1)<=8#?

1 Answer
smendyka
Jul 23, 2017

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.

#-8 <= 3d - 1 <= 8#

First, add #color(red)(1)# to each segment of the system of inequalities to isolate the #d# term while keeping the system balanced:

#-8 + color(red)(1) <= 3d - 1 + color(red)(1) <= 8 + color(red)(1)#

#-7 <= 3d - 0 <= 9#

#-7 <= 3d <= 9#

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

#-7/color(red)(3) <= (3d)/color(red)(3) <= 9/color(red)(3)#

#-7/3 <= (color(red)(cancel(color(black)(3)))d)/cancel(color(red)(3)) <= 3#

#-7/3 <= d <= 3#

Or

#d >= -7/3# and #d <= 3#

Or, in interval notation:

#[-7/3, 3]#

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