Question

What is the maximum number of solutions an

equation of the form ||ax − b| + c| = d can have?
Justify your reasoning with an example

Answer 1

• 4 for c < 0
• 2 for c > 0

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.