Answer:
Step-by-step explanation:
You want the maximum possible number of solutions to the equation
||ax -b| +c| = d
Analysis
The values of 'a' and 'b' represent a vertical scale factor and a horizontal shift, so don't really contribute to the number of possible solutions. That means we can consider only the equation ...
||x| +c| = d
Positive c
We know |x| is always positive, and may have the same value for two different values of x. For positive 'c', the outside absolute value function does nothing, so the equation is effectively ...
|x| = d-c
For d < c, there can be no solutions. For d = c, there will be one solution, and for d > c, there will be two solutions.
Negative c
When 'c' is negative, then sum |x| +c may have either sign. This gives rise to another pair of possible solutions, for a total of four solutions.
Graph
The attached graph shows the two cases. The red graph shows the left side expression with c > 0; the purple graph shows the left side expression with c < 0. The orange horizontal line represents a value of 'd'. The number of intersection points with that horizontal line is the number of possible solutions to the given equation.
For c > 0, there are two possible solutions.
For c < 0, there are four possible solutions.
