Example 7. Solve the equation

$$
|x-| 4-x||-2 x=4
$$
Solution. The given equation is equivalent to the combination of two systems:

$$
\left\{\begin{array} { l } 
{ 4 - x \geqslant 0 } \\
{ | x - ( 4 - x ) | - 2 x = 4 , }
\end{array} \quad \left\{\begin{array}{l}
4-x4 \\
-2 x=0
\end{array}\right.\right.
$$

The second system of the combination (4) has no solutions.

The first system of the combination (4) is equivalent to the combination of the following two systems:

$$
\left\{\begin{array} { l } 
{ x \leqslant 4 , } \\
{ 2 x - 4 \geqslant 0 , } \\
{ ( 2 x - 4 ) - 2 x = 4 , }
\end{array} \quad \left\{\begin{array}{l}
x \leqslant 4 \\
2 x-4<0 \\
-(2 x-4)-2 x=4
\end{array}\right.\right.
$$

i.e., the combination

![](https://cdn.mathpix.com/cropped/2024_05_21_aca6fef227914c05e0acg-039.jpg?height=179&width=562&top_left_y=2526&top_left_x=744)

The only solution to the combination (5), and therefore to the original equation, is the number 0.

Consider an equation of the form

$$
\left|f_{1}(x)\right|+\left|f_{2}(x)\right|+\left|f_{3}(x)\right|+\ldots+\left|f_{n}(x)\right|=g(x)
$$

where \( f_{1}(x), \ldots, f_{n}(x), g(x) \) are some functions. If this equation is solved by sequentially opening the absolute value signs, then after opening one absolute value sign, a combination of two systems is obtained, after opening the second absolute value sign, a combination of four systems, and so on. This method is very cumbersome. Such equations are easier to solve using the interval method. For this, first find all points where at least one of the functions \( f_{1}(x), f_{2}(x), \ldots, f_{n}(x) \) changes sign. These points divide the domain of admissible values of equation (6) into intervals, on each of which all functions \( f_{1}(x), f_{2}(x), \ldots, f_{n}(x) \) retain their sign. Then, using the definition of absolute value, transition

![](https://cdn.mathpix.com/cropped/2024_05_21_aca6fef227914c05e0acg-040.jpg?height=323&width=1079&top_left_y=1118&top_left_x=497)

Fig. 2.2

from equation (6) to a combination of systems without absolute value signs.