A rational equation means that you are setting two rational expressions equal to each other. The goal is to solve for x; that is, find the value(s) that make the equation true.
Suppose you are told that:
If you think about it, the x in this equation has to be a 3. That is to say, if
This leads us to a very general rule: If you have a rational equation where the denominators are the same, then the numerators must be the same.
This in turn suggests a strategy: find a common denominator, and then set the numerators equal.
For example, consider the rational equation
by factoring the denominators,we find that we must multiply the left side of the equation by
Based on the rule above—since the denominators are equal, we can now assume the numerators are equal, so we know that
What we’re dealing with, in this case, is a quadratic equation. As always, move everything to one side, giving
and then factor. A common mistake in this kind of problem is to divide both sides by x; this loses one of the two solutions.
Two solutions to the quadratic equation. However, in this case,
As always, it is vital to remember what we have found here. We started with the equation