Double absolute value equations.

I'm self-studying Cambridge IGCSE additional maths and I've got to the double absolute value equations but I've been really struggling to understand how it works for a while now.

My textbook gives an example of an equation and how to solve it as shown below.

$\left | x+4 \right | +\left | x-5 \right | = 11 $

$\left | x+4 \right |= 11 -\left | x-5 \right | $

$x+4 = 11 -\left | x-5 \right |$       (1)
$x+4 = \left | x-5 \right |-11$       (2)

Using equation (1)
$\left | x-5 \right | = 7-x$   
$ x-5 = 7-x$   
$2x = 12$

$0=-2$ Which is false

Using equation (2)
$\left | x-5 \right |=x+15$
$0=20$ which is false

$ x-5=-(x+15)$

The solution is $x=6$ or $x=-5$

I know how to solve an equation like

$\left | x+5 \right |= \left | x-4 \right |$

I just don't understand what is happening here. Could you help explain it step by step why each step is taken and if possible could you graph it so that I can visually see what is happening?

Thanks for the help.


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Mathe Mathe
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