A fraction is defined as a number that can be written in the form $\frac{p}{q}$ where $p$ and $q$ are integers and $q\neq0$.
In the same way, rational expressions can be defined as an expression which can be written in the form $\frac{p}{q}$ where $p$ and $q$ are polynomials and $q\neq0$.
All the operations that are used in fractions can also be performed in rational expression.
Rational expression can be simplified such that it can't be reduced further. There are different ways to simplify it. The basic steps involves changing the polynomials of both numerators and denominator into factors.
To solve a rational expression means we are trying to solve a rational equation. It means we are trying to solve the variable in the rational expression. This value will satisfy the equation.
Lets consider an example
Solve for x,
Divide both side by $x-3$
$\frac{(x^{3}+10x^{2}+24x)}{(x-3)}\times{(x-3)}=0\times {(x-3)}$
It can be seen that x is common to all terms.
So we get two factors $x$ and $(x^{2}+10x+24)$
Using the equation $x=0$, we get $x=0$ as the solution.
Now, lets solve for $(x^{2}+10x+24)=0$
This is a quadratic equation which can be either solved by factoring or by using quadratic formula.
Lets use quadratic formula to solve it.
First compare it to the standard quadratic equation $ax^{2}+bx+c=0$
So $a=1$, $b=10$ and $c=24$
By the quadratic formula, the solution for
$x=\frac{-b\pm \sqrt{b^{2}-4ac}}{2a}$
    =$\frac{-10\pm \sqrt{(-10)^{2}-4\times 1\times 24}}{2\times 1}$
     =$\frac{-10\pm \sqrt{100-96}}{2}$
      =$\frac{-10\pm \sqrt{4}}{2}$
       =$\frac{-10\pm 2}{2}$
so, we get $x=\frac{-10+2}{2}$ and $x=\frac{-10-2}{2}$
                    = $\frac{-8}{2}$ and        =$\frac{-12}{2}$
                    = $-4$          and       =$-6$
So the solution of the rational expressions are $0$, $-4$ and $-6$