Is this equation solvable (by hand) and what do you call this equation?

Question

$a = x - \min(b\, x, c)$

$a, b, c$ are known. We need to solve for $x$.

If it is, and I can get guidance as to what this is called, and where there are resources to find out how to solve, this, I'm happy to do this.

You simply test each condition: $a=x-bx \text{if} bx<c$ and $a=x-c$ if $bx>c$. For equality choose either of the condtions. Solve for $x$ and test which condition holds. — player100, Sep 04 '18 at 12:58

score 1 · Accepted Answer · answered Sep 04 '18 at 13:41

Expanding on the comment by player100:

The function $f(x) = x-\min(bx,c) $ has two regions:

$f(x) = (1-b)x \quad \text{if } bx \le c$

$f(x) = x-c \quad \text{if } bx \gt c$

Setting $f(x) = a$ then we have

$x = \frac{a}{1-b} \quad \text{if } \frac{ab}{1-b} \le c$

$x = c+a \quad \text{if } b(c+a) \gt c$

Note that we can re-arrange the condition $b(c+a) \gt b$ as $ab \gt (1-b)c$, so the two conditions are mutually exclusive if $1-b > 0$ i.e. $b < 1$. So if $b< 1$ then the equation $f(x)=a$ has exactly one solution.

However, if $b > 1$ then the equation $f(x)=a$ will have two solutions if $ab > (1-b)c$, one solution if $ab = (1-b)c$ and no solutions if $ab < (1-b)c$.

Thank you very much for your help and solution! – RyanH Sep 06 '18 at 10:15 — RyanH, Sep 06 '18 at 10:15

Is this equation solvable (by hand) and what do you call this equation?

1 Answers1