Question : Find x values which satisfies the following equation

Question

2|x| > |2x -5| + |x+1|

I squared all the terms in both sides of the = sign to get (x+4)^2 + 10 < 0

Where have I made a mistake as I'm getting imaginary values for x?

Answer 1

squaring is something we don't normally do with inequalities,
btw, how did you come up with that result ??

This type of question can be quite tedious, and you have to be meticulous in your work.
Let's start with:
|2x -5| < 2|x| - |x+1|
this splits into ...
2x - 5 < 2|x| - |x+1| AND 2x-5 > -2|x| + |x+1|

I will label 2x - 5 < 2|x| - |x+1| as *
and 2x-5 > -2|x| + |x+1| as **

working on *
2x - 5 < 2|x| - |x+1|
|x+1| < 2|x| - 2x + 5 <------ same as *

x+1 < 2|x| - 2x + 5 <---- call this *1
and
x+1 > -2|x| + 2x - 5 <---- call this *2

working on *1
2|x| > 3x - 4 , this leads to ...
2x > 3x - 4 OR 2x < -3x + 4
-x > -4 OR 5x < 4
x < 4 OR x < -4/5
x < 4

back to *2
x+1 > -2|x| + 2x - 5
2|x| > 2x - 5 -x -1
2|x| > x - 6 , which leads to
2x > x-6 OR 2x < -x + 6
x > -6 OR 3x < 6
x > -6 OR x < 2 which would be all values of x

BUT *1 AND *2
so x < 4 <------ result of *

so that's only half the work, we still need **

go for it

final result should be
2 < x < 4

check:
www.wolframalpha.com/input/?i=2%7Cx%7C+%3E+%7C2x+-5%7C+%2B+%7Cx%2B1%7C

perhaps some of the other math tutors has an easier way

Answer 2

|2x| > |2x-5| + |x+1|.
2x > 2x-5 + x+1.
2x -2x - x > -4,
-x > -4,
X < 4. Reversed inequality sign after dividing by -1.