To evaluate a rational expression at a particular value of each variable, evaluate the numerator and denominator and divide the results. The rational expression is undefined whenever the bottom evaluates to 0.
We add, subtract, multiply, and divide rational expressions using the same techniques as for rational numbers. But systematic factoring is now more important, since less can be done by trial and error. When multiplying or dividing, you should factor all of the numerators and denominators, and there is no need to get common denominators; when adding or subtracting, you only need to factor the denominators at first, and you need to make all of these denominators the same. (You should still factor the numerator of the final answer when adding or subtracting, but that's only to simplify the result.)
A fraction whose top or bottom (or both) contains fractions is called complex. The straightforward way to simplify a complex fraction is to simplify the top and bottom separately and then divide them. Another way is to find a common denominator, not of just the fractions in the top or of just the fractions in the bottom, but of all of the fractions that appear in either; then you can multiply both top and bottom by this common denominator to turn the complex fraction into a simple one. (You'll still need to simplify this fraction, however.)
There is another method of solving rational equations. Here, you just factor the denominators, then multiply both sides of the equation by a common denominator. This is especially helpful when the equation has addition or subtraction of rational expressions; instead of simplifying each side (which also requires finding common denominators) you can multiply everything by a common denominator and make all of the fractions go away. (You still have to solve the resulting polynomial equation.)
Either way, you must check for extraneous solutions! Since every method of solving rational equations involves multiplying both sides by something that might be zero, you need to check that none of these expressions that you multiplied by actually evaluates to zero for any of your solutions. Equivalently, you can check that the original expressions in the equation are both defined for all of your solutions, with no division by zero. If any of your solutions fails this check, then you must throw it out. (Sometimes you'll throw out all of them, sometimes none of them; you never know until you check.)
This web page was written between 2010 and 2016 by Toby Bartels, last edited on 2016 August 2. Toby reserves no legal rights to it.
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