# 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)
$\left | x-5 \right | = 7-x$
$x-5 = 7-x$
$2x = 12$
$x=6$

Or
$x-5=-(7-x)$
$0=-2$ Which is false

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

or
$x-5=-(x+15)$
$2x=-10$
$x=-5$

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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