Answer
a) One solution
b) Two solutions
Work Step by Step
a) When $c = d$ and $a>0$ $$a | x + b | + c = d$$ becomes $$a | x + b | + d = d$$ therefore after subtracting $d$ from both sides $$a | x + b | = 0$$ and after dividing both sides by $a$ $$ |x + b| = 0.$$ This equation has the right hand side equal to $0$, which will always have one solution.
b) $a<0$ and $c>d$
$$a | x + b | + c = d$$ becomes after subtracting $d$ from both sides (lets say $d - c = -k<0$ since $c>d$) $$ a| x + b | = -k.$$ Dividing both sides by $a$, let's say $\frac{k}{a} = v$, where $v>0$ since $a <0$ and $k<0$ $$| x + b | = v.$$ As $ v>0$, this equation will have $2$ solutions.
