Rational equations

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.)

To solve an equation between rational expressions, you can cross-multiply: multiply the top of one side by the bottom of the other side, then set the two resulting polynomials equal. In other words, to solve A/B = C/D, you solve AD = BC, or equivalently solve AD − BC = 0. If you are unable to factor this polynomial, then try factoring the original tops and bottoms; anything that shows up in both A or D and in B or C can then be factored out.

There is another method of solving rational equations, which corresponds to the second method of simplifying complex fractions. 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.)

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