Question : Find x values which satisfies the following equation

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?

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

To find the values of x that satisfy the equation 2|x| > |2x - 5| + |x + 1|, squaring all the terms in both sides of the equation is not a correct approach. Squaring an inequality can introduce extraneous solutions, which are solutions that do not actually satisfy the original inequality.

Here's the correct way to solve the equation:

First, let's split the absolute value expressions into two cases based on the sign of the expression inside the absolute value.

Case 1: (2x - 5) is positive or zero.
In this case, the equation becomes:
2x > (2x - 5) + (x + 1)

Simplifying this equation:
2x > 2x - 5 + x + 1
2x > 3x - 4

Rearranging the terms:
2x - 3x > -4
-x > -4

Dividing both sides of the inequality by -1 and flipping the inequality sign:
x < 4

Case 2: (2x - 5) is negative.
In this case, the equation becomes:
2x > -(2x - 5) + (x + 1)

Simplifying this equation:
2x > -2x + 5 + x + 1
2x > -x + 6

Rearranging the terms:
2x + x > 6
3x > 6

Dividing both sides of the inequality by 3:
x > 2

So, we have two sets of solutions:
1. x < 4
2. x > 2

To find the intersection of these two sets, we take the values of x that satisfy both conditions.

Therefore, the values of x that satisfy the original equation are:
x < 4 and x > 2

This can be written as 2 < x < 4.

|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.