Is a uniformly continuous function vanishing at $0$ bounded by $a|x|+c$?
It is false if $c=0$. To see this, try to think of a continuous function that grows very rapidly near $0$.
It is true if $c\gt 0$. One way to show it is by taking a number of very small steps from $0$ to $x$, small enough to guarantee (using uniform continuity) that the function changes no more than a certain fixed amount at each step. Trying to write out the details should lead you to what this fixed amount is, and to what value of $a$ will work.