Largest Value of a for Infinite Solutions

The equation $|x - u| + |x - v| = d$ has infinitely many solutions if and only if $d = |u - v|$, and in that case the solutions are all $x$ in the interval between $u$ and $v$.

Here, $u = -a$ and $v = 1$. So we must have $|-a - 1| = 2$, which means $|a + 1| = 2$.

Thus: $a + 1 = 2 \implies a = 1$ $a + 1 = -2 \implies a = -3$

The largest real value of $a$ is $1$.