Hey, ! In my pre-calculus class the teacher showed the solution of the following example: \begin{align} \vert x-3 \vert \lt \vert x - 4 \vert + x \end{align} He started by stated the domains needed to be checked:
\begin{aligned} \lbrack 4, +\infty ) \newline \lbrack 3, 4 ) \newline ( -\infty, 3) \end{aligned}
Which I don't have a based lead on how he come to these domains and then deducted the following inequalities ( for each domain respectively): \begin{align} x-3 \lt x - 4 + x \newline x-3 \lt -(x-4) + x \newline -(x-3) \lt -(x-4) + x \end{align}
The final solution was \begin{aligned} (-1, +\infty ) \end{aligned}
Now I cannot understand how he deducted the domains and the right inequalities(there is one missing possibility): \begin{align} -(x-3) \lt x-4 + x \end{align}
It may be apparent but I still cannot wrap my mind around it. Thanks!
In my answer here, I describe a method for solving multiple-absolute-value equations that can similarly be applied to inequalities. It is the same idea as what your teacher showed and what's discussed in the comments, but the particular organization of the work by dividing up a number line and working under each section may be helpful.