Example 9. Solve the equation

$$
|x|+|7-x|+2|x-2|=4
$$
Solution. Using the interval method, we find (Fig. 2.3) the intervals of sign constancy for the expressions \(x, 7-x\), and \(x-2: x7\).

![](https://cdn.mathpix.com/cropped/2024_05_21_aca6fef227914c05e0acg-041.jpg?height=409&width=970&top_left_y=612&top_left_x=546)

Fig. 2.3

Thus,

![](https://cdn.mathpix.com/cropped/2024_05_21_aca6fef227914c05e0acg-041.jpg?height=439&width=854&top_left_y=1168&top_left_x=601)

The first system of the conjunction is equivalent to the system

\[
\left\{\begin{array}{l}
x \leqslant 0 \\
x=7 / 4
\end{array}\right.
\]

and, therefore, has no solutions.

The second system of the conjunction is equivalent to the system

\[
\left\{\begin{array}{l}
07 \\
x=15 / 4
\end{array}\right.
\]

and, therefore, has no solutions.

Thus, the conjunction ( \(8^{\prime}\) ), and hence the original equation, has no solutions.

Remark. The solutions to equations of the form

\[
\begin{aligned}
& |x-a|=c \\
& |x-a|+|x-b|=c \\
& |x-a|-|x-b|=c
\end{aligned}
\]

(given numbers \(a, b, c>0\)) admit a simple geometric interpretation.

Solving the equation (9) - means finding all points on the number line \(O x\) that are at a distance \(c\) from the point with coordinate \(a\). There are two such points: the point with coordinate \((c+a)\) and the point with coordinate \((a-c)\).

Solving the equation

\[
|x-1|+|x-3|=6
\]

- means finding all such points on the number line \(O x\) for each of which the sum of the distances from it to the points with coordinates 1 and 3 is 6. Clearly, no point on the segment \([1 ; 3]\) satisfies this condition, as the sum of the specified distances for any of them is 2 (i.e., not 6). Outside this segment, there are only two such points: the point with coordinate 5 and the point with coordinate \((-1)\).

Similarly, the solution to an equation of the form (11) can be interpreted. For example, to solve the equation

\[
|x-1|-|x-3|=2
\]

we need to find all such points on the number line \(O x\) for each of which the difference between the distance from it to the point with coordinate 1 and the distance from it to the point with coordinate 3 is 2. Since the length of the segment \([1 ; 3]\) is 2, it is clear that any point with coordinate \(x \geqslant 3\) satisfies, while any point with coordinate \(x<3\) does not satisfy it. Thus, the solution to the original equation is the set of all numbers from the interval \([3 ;+\infty)\).