15) How many real solutions does the equation

$$
|||a|+3|-2|=1 ?
$$

(A) None,
(B) one,
(C) two,
(D) three, (E) eight.
15. The answer is $(\mathbf{B})$.

||$|a|+3|-2|=1$ means that ||$a|+3|-2$ can be equal to 1 or equal to -1, that is

$|a|+3 \mid$ is 3 or 1. Regardless of the value of $a$, its absolute value is non-negative,

so ||$a|+3| \geq 3$ in any case. Consequently, the only possibility is that ||$a|+3|=3$, from which $a=0$. The equation has only one solution.