question on the diffrence of rational and solving an equatio

[nae]

could someone please explain to me what the diffrence is between adding two rational expressions and solving an equation?

 

[pka]

An expression is just that. For example: $\displaystyle \frac{1}{{x^2 - 1}} + \frac{{ - x}}{{\left( {x + 1} \right) }}$.
That example contains no equal sign. There is nothing to solve because there is no ‘missing’ information. Now we can do operations of that expression. We can combine the two fractions into one. But that is not solving anything, it is just operation.

On the other hand $\displaystyle \frac{1}{{x^2 - 1}} + \frac{{ - x}}{{\left( {x + 1} \right)}} = 1$ is an equation.
It states that the rational expression on the left equals 1 for certain unknown values of x. To SOLVE this equation is to find the values of x that make the equation TRUE. To do the solving we will combine the left side and proceed. We can show that there are two values of x, $\displaystyle \frac{{ - 1 \pm \sqrt {17} }}{4}$ that work.