smc

Let $A$ be point $3$ on a number line, and let $B$ be point $4$. Let $X$ be a mobile point at $x$. Geometrically, $|x-4|+|x-3|$ represents $AX+BX$. If $X$ is between $A$ and $B$, then $AX+BX=AB=1$. Otherwise, if $X$ is to the left of $A$, then $AX+BX=AX+(AX+AB)=2AX+1$, which is greater than $1$. If $X$ is to the right of $B$, we have $AX+BX=(AB+BX)+BX=2BX+1$. In all $3$ cases, the minimum value of $AX+BX$ is $1$. Thus, $|x-4|+|x-3|<a$ will always be true if $a>1$. If $a=1$, it can be false for $3\le x\le 4$. If $a<1$, then $|x-4|+|x-3|<a$ is always false because the LHS is too big. Thus, the answer is $\boxed{a>1}$.