  In this work we analyze traversable wormhole solutions in the linear form of
$f\left(R,T\right)=R+\lambda T$ gravity satisfying the Null, Weak, Strong, and
Dominant Energy Conditions (NEC, WEC, SEC, and DEC respectively) for the entire
spacetime. These solutions are obtained via a fully analytical parameter space
analysis of the free parameters of the wormhole model, namely the exponents
controlling the degree of the redshift and shape functions, the radius of the
wormhole throat $r_0$, the value of the redshift function at the throat
$\zeta_0$, and the coupling parameter $\lambda$. Bounds on these free
parameters for which the energy conditions are satisfied for the entire
spacetime are deduced and two explicit solutions are provided. Even if some of
these bounds are violated, leading to the violation of the NEC at some critical
radius $r_c>r_0$, it is still possible to find physically relevant wormhole
solutions via a matching with an exterior vacuum spacetime in the region where
the energy conditions are still satisfied. For this purpose, we deduce the set
of junction conditions for the form of $f\left(R,T\right)$ considered and
provide an explicit example. These results seem to indicate that a wide variety
of non-exotic wormhole solutions are attainable in the $f\left(R,T\right)$
theory without the requirement of fine-tuning.
