Consider equations involving absolute values  |  a x + b |  and  | c x + d |.
To solve a similar equation, it is necessary to find intervals, on each of which expressions  a x + b   and  c x + d  conserve their signs, and so the absolute values symbols can be dropped each time, provided that the correct signs are written in front of the expressions.
As a result, the given problem is reduced to solving a few ordinary equations.

Let  x = x1  and  x = x2  be the solutions of the equations,

a x + b = 0

c x + d = 0.

Points  x1  and  x2  divide the number line into three intervals shown in the figure below,

where  x1  is assumed to be less than  x2.

Then we need to consider three cases,

solving each time an equation without any absolute values.

The solution set of the given equation is the union of solutions obtained in all cases.

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