How do you solve #abs(3x-9)-2<=7#?

1 Answer
Stefan V.
Jul 22, 2016

#x in [0, 6]#

Explanation:

The first thing to do here is isolate the modulus on one side of the equation by adding #2# to both sides

#|3x-9| - color(red)(cancel(color(black)(2))) + color(red)(cancel(color(black)(2))) <= 7 + 2#

#|3x - 9| <= 9#

Now, you have two possible cases to deal with

  • #3x - 9 >=0 implies | 3x-9| = 3x-9#

In this case, the inequality becomes

#3x-9 <= 9#

#3x <= 18 implies x <= 18/3 = 6#

  • #3x - 9 < 0 implies |3x - 9| = - (3x-9)#

In this case, the inequality becomes

#-(3x-9) <= 9#

#-3x +9 <=9#

#-3x <= 0 implies x >= 0#

You thus know that any value of #x# that is smaller than or equal to #6# and greater than or equal to #0# will satisfy the original inequality. In interval notation, this is written as

#x in (-oo, 6] nn [0, + oo)#

This means that the solution interval will be

#x in [0, 6]#

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