How do you solve #abs(3x-8)=x#?

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Answer 1 · 2018-03-24T10:09:59

The solutions are #S={2, 4}#

Let's rewrite the equation

#|3x-8|-x=0#

#|3x-8|=3x-8#, if #3x-8>=0#

and

#|3x-8|=-(3x-8)=-3x+8#, if #3x-8<0#

Let #f(x)=|3x-8|-x=0#

Let's build a variation table

#color(white)(aaaa)##x##color(white)(aaaaaaaa)##-oo##color(white)(aaaaaaaa)##0##color(white)(aaaaaaaaaa)##8/3##color(white)(aaaaaaa)##+oo#

#color(white)(aaaa)##x##color(white)(aaaaaaaaaaaaa)##-##color(white)(aaaa)####color(white)(aaaaa)##+##color(white)(aaaaaaaa)##+#

#color(white)(aaaa)##|3x-8|##color(white)(aaaaaa)##-3x+8##color(white)(a)####color(white)(aaa)##-3x+8##color(white)(aaaaa)##3x-8#

#color(white)(aaaa)##f(x)##color(white)(aaaaaaaaaaa)##x=2##color(white)(a)####color(white)(aaaaa)##x=2##color(white)(aaaaaa)##x=4#

The solutions are #S={2, 4}#

graph{|3x-8|-x [-7.9, 7.904, -3.95, 3.95]}