How do you solve # abs(2 - 5x )>= 2 / 3#?

1 Answer
Stefan V.
Aug 19, 2017

#x in (-oo, 4/15] uu [8/15, + oo)#

Explanation:

You're dealing with an absolute value inequality, so right from the start, you should know that you must look at two possible cases

#color(white)(a)#

  • #2 - 5x >= 0 implies |2 - 5x| = 2 - 5x#

In this case, you have

#2 - 5x >= 2/3#

This will get you

#-5x >= 2/3 - 2#

#-5x >= -4/3 implies x <= 4/15#

#color(white)(a)#

  • #2 - 5x < 0 implies |2 - 5x| = - (2 - 5x)#

In this case, you have

#-2 + 5x >= 2/3#

This will get you

#5x >= 2/3 + 2#

#5x >= 8/3 implies x >= 8/15#

#color(white)(a)#

You can thus say that the solution interval for the original inequality will be

#x in (-oo, 4/15] uu [8/15, + oo)#

This tells you that the original inequality is satisfied if #x# is less than or equal to #4/15# or greater than or equal to #8/15#. In other words, any value of

#x in (4/15, 8/15)#

is not a solution to the original inequality.

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