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!