Example 8. Solve the equation

$$
|3 x-8|-|3 x-2|=6
$$
Solution. Using the interval method (Fig. 2.2), we find the intervals of sign constancy of the expressions $3 x-8$ and $3 x-2: x8 / 3$.

Thus, equation (7) is equivalent to the union of three systems:

$$
\begin{aligned}
& \left\{\begin{array} { l } 
{ x \leqslant 2 / 3 , } \\
{ - ( 3 x - 8 ) + ( 3 x - 2 ) = 6 , }
\end{array} \quad \left\{\begin{array}{l}
2 / 3 < x \leqslant 8 / 3 , \\
(3 x-8) + (3 x-2) = 6 , 
\end{array} \quad \left\{\begin{array}{l}
x > 8 / 3 , \\
(3 x-8)-(3 x-2)=6
\end{array}\right.
\end{aligned}
$$

i.e., the union of the systems

$$
\left\{\begin{array} { l } 
{ x \leqslant 2 / 3 , } \\
{ 6 = 6 , }
\end{array} \quad \left\{\begin{array} { l } 
{ 2 / 3 < x \leqslant 8 / 3 , } \\
{ 6 x - 10 = 6 , }
\end{array} \quad \left\{\begin{array} { l } 
{ x > 8 / 3 , } \\
{ -6=6 }
\end{array}\right.\right.\right.
$$

The solution to the first system is all numbers from the interval $(-\infty ; 2 / 3]$. The second and third systems have no solutions.

Thus, the set of all solutions to the original equation is the interval $(-\infty ; 2 / 3]$.