Answer
$\{-\frac{1}{2},3\}$.
Work Step by Step
First we determine the Least Common Denominator (LCD) so that we clear the fractions. The LCD is $3x^2$.
Multiply the equation by $3x^2$.
$3x^2 \cdot \left (\frac{2}{3}-\frac{5}{3x}\right )=3x^2 \cdot\left ( \frac{1}{x^2}\right )$
Use the distributive property.
$3x^2 \cdot \frac{2}{3}-3x^2 \cdot \frac{5}{3x}=3x^2 \cdot \frac{1}{x^2}$
Simplify.
$2x^2 -5x =3$
Add $-3$ to both sides.
$2x^2 -5x-3 =3-3$
Simplify.
$2x^2 -5x-3 =0$
Rewrite the middle term $-5x$ as $-6x+1x$.
$2x^2 -6x+1x-3 =0$
Group the terms.
$(2x^2 -6x)+(1x-3) =0$
Factor each term.
$2x(x -3)+1(x-3) =0$
Factor out $(x-3)$.
$(x -3)(2x+1) =0$
Set each factor equal to zero.
$x -3=0$ or $2x+1 =0$
Isolate $x$.
$x =3$ or $x=-\frac{1}{2}$
The solution set is $\{-\frac{1}{2},3\}$.
Note: Check if the solution is correct. The equation is defined for all real values of $x$ except the zeros of the denominators, which is $o$. Since our solution does not contain $x=0$, it is correct.
