Answer
$\left\{-\dfrac{5}{2},3\right\}$
Work Step by Step
Multiplying both sides by the $LCD=
x
,$ the given equation, $
\dfrac{15}{x}=2x-1
$, is equivalent to
\begin{align*}
x\left(\dfrac{15}{x}\right)&=(2x-1)x
\\\\
1(15)&=(2x)x-1(x)
\\
15&=2x^2-x
\\
0&=2x^2-x-15
\\
2x^2-x-15&=0
.\end{align*}
Using factoring of trinomials, the equation above is equivalent to
\begin{align*}
(x-3)(2x+5)&=0
.\end{align*}
Equating each factor to zero (Zero Product Property) and solving for the variable, then
\begin{array}{l|r}
x-3=0 & 2x+5=0
\\
x=3 & 2x=-5
\\
& x=-\dfrac{5}{2}
.\end{array}
Hence, the solution set of the equation $
\dfrac{15}{x}=2x-1
$ is $
\left\{-\dfrac{5}{2},3\right\}
$.
