Nonlinear System

$\begin{aligned} \color{#3D99F6}{ab} \times \color{#D61F06}{ca} & = \color{#3D99F6}{2} \times \color{#D61F06}{4} \\ a^2\color{#20A900}{bc} & = 8 \\ \color{#20A900}{3}a^2 & = 8 \\ \implies a^2 & = \frac 83 \\ a & = \pm \sqrt{\frac 83} \end{aligned}$

$\begin{aligned} |a+\color{#3D99F6}{b}+\color{#D61F06}{c}| & = \left| a+\color{#3D99F6}{\frac 2a}+\color{#D61F06}{\frac 4a} \right| \\ & = \left| a+\frac 6a \right| \\ & = \left| \pm \sqrt{\frac 83} \pm 6 \sqrt{\frac 38} \right| \\ & \approx \boxed{5.3} \end{aligned}$

$ab = 2 \\ b = \dfrac{2}{a} \\ bc = 3\\ \dfrac{2}{a}c = 3 \\ c = \dfrac{3a}{2} \\ ca = 4 \\ \dfrac{3a}{2}a = 4 \\ a^2 = \dfrac{8}{3} \\ a = \pm \sqrt[]{\dfrac{8}{3}} = \pm 2\sqrt[]{\dfrac{2}{3}} \\ b = \dfrac{2}{a} = \dfrac{2}{\pm 2\sqrt[]{\dfrac{2}{3}}} = \pm \sqrt[]{\dfrac{3}{2}} \\ c = \dfrac{3a}{2} = \dfrac{\pm3\times 2\sqrt[]{\dfrac{2}{3}}}{2} = \pm 3\sqrt[]{\dfrac{2}{3}} \\ \boxed{\therefore |a+b+c| = |\pm 2\sqrt[]{\dfrac{2}{3}} + \pm \sqrt[]{\dfrac{3}{2}} + \pm 3\sqrt[]{\dfrac{2}{3}}| = \\ 5\sqrt[]{\dfrac{2}{3}} + \sqrt[]{\dfrac{3}{2}} = \dfrac{13}{\sqrt[]{6}} \approx 5.3072}$

ab = 2 (i)

bc = 3 (ii)

ca = 4 (i)

It is easy to see, that either all three variables (a, b, c) are positive or they are all negative.

(i) × (ii) × (iii) :

$a^2b^2c^2 = 24$

$abc = ± \sqrt {24} = 2 \sqrt {6} \ \text { (iv) }$

$a = \frac {abc}{bc} = \frac { ± 2 \sqrt {6} }{3}$

$b = \frac {abc}{ca} = \frac { ± 2 \sqrt {6} }{4}$

$c = \frac {abc}{ab} = \frac { ± 2 \sqrt {6} }{2}$

Therefore

$| a + b + c | = | ± 2 \sqrt {6} ( \frac {1}{3} +\frac {1}{4} + \frac {1}{2}) | = | ± 2 \sqrt {6} × \frac {13}{12} |$

$= \frac {13 \sqrt {6} }{6} = \boxed {5.3} \text { (1 d. p.) }$