.
how many solution does the equation a|x+b|+c=d have if a < 0 and c = d? If a < 0, d > 0 and c < d ? Explain
~~~~~~~~~~~~~~~~~~~~
(a) How many solution does the equation a|x+b|+c=d have if a < 0 and c = d ?
An equation
a|x+b| + c = d
is equivalent to
a|x+b| = d - c.
If c = d, then this equation becomes
a|x+b| = 0.
If a =/= 0 (as it is given in our case), then the last equation implies
|x+b| = 0,
which has one and only one unique solution x= -b.
Answer. In this case the original equation has a unique solution.
(b) How many solution does the equation a|x+b|+c=d have if a < 0, d > 0 and c < d ?
An equation
a|x+b| + c = d
is equivalent to
a|x+b| = d - c. (*)
If c < d, then the right side of the last equation is positive, while its left side is negative.
Therefore, the equality (*) is not possible.
Or, in other words, the equation (*) has no solutions.
Answer. Under the given condition, the given equation has no solution.
Solved, answered and explained.