The values of x for which the functions f(x) = x – 3 and $\mathrm{\varphi }\left(\mathrm{x}\right)=4-\mathrm{x}$ satisfy the inequality $\left|\mathrm{f}\right(\mathrm{x})+\mathrm{\varphi }(\mathrm{x}\left)\right|<\left|\mathrm{f}\right(\mathrm{x}\left)\right|+\left|\mathrm{\varphi }\right(\mathrm{x}\left)\right|$ are

We have $\mathrm{f}\left(\mathrm{x}\right)+\mathrm{\varphi }\left(\mathrm{x}\right)=\mathrm{x}-3+4-\mathrm{x}=1$, so that

$\left|\mathrm{f}\right(\mathrm{x}\left)\right|+\left|\mathrm{\varphi }\right(\mathrm{x}\left)\right|=1$

Furthermore,

$\left|\mathrm{f}\right(\mathrm{x}\left)\right|=\left\{\begin{array}{l}\mathrm{x}-3\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{ if }\mathrm{x}\ge 3\\ 3-\mathrm{x}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{ if }\mathrm{x}<3\end{array}\right.$;

and $\left|\mathrm{\varphi }\right(\mathrm{x}\left)\right|=\left\{\begin{array}{l}4-\mathrm{x}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{ if }\mathrm{x}\le 4\\ \mathrm{x}-4\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{ if }\mathrm{x}>4\end{array}\right.$

$\Rightarrow \left|\mathrm{f}\right(\mathrm{x}\left)\right|+\left|\mathrm{\varphi }\right(\mathrm{x}\left)\right|=\left\{\begin{array}{cl}7-2\mathrm{x}& \text{ if }\mathrm{x}<3\\ 1& \text{ if }3\le \mathrm{x}\le 4\\ 2\mathrm{x}-7& \text{ if }\mathrm{x}>4\end{array}\right.$

We need those points for which the L.H.S. is greater than 1. Clearly, we can exclude values of x between 3 and 4. Now, for values of x less than 3, 7 – 2x is greater than 1, and for values of x greater than 4, 2x – 7 is greater than 1.

Therefore, the given inequality is true for values of x given by $(-\mathrm{\infty }, \mathrm{\infty })$ – [3, 4].