Review question

# When can this equation involving algebraic fractions hold? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R8407

## Suggestion

1. Show that, if $a$ and $b$ are either both positive or both negative, then the equation $\frac{x}{x-a}+\frac{x}{x-b}=1$ has two distinct real solutions.

How can we handle equations which involve fractions? Does it make a difference if $x$ is in the denominator?

1. Show that the equation $\frac{x}{x-a}+\frac{x}{x-b}=1+c$ has exactly one real solution if $c^2=-\dfrac{4ab}{(a-b)^2}$. Show that this equation can be written $c^2=1-\left(\dfrac{a+b}{a-b}\right)^2$ and deduce that it can only hold if $0 < c^2 \leq 1$.

This interactive applet shows the graph of $y = \dfrac{x}{x-a}+\dfrac{x}{x-b}-1$. You can change the values of $a$ and $b$.

The blue lines are the lines $y=c$ which intersect the red curve exactly once.