Rational equations (§§7.7&7.8)

Recall that a fraction whose top or bottom (or both) contains fractions is called complex. The straightforward way to simplify a complex fraction (Method Ⅰ in the textbook) is to simplify the top and bottom separately and then divide them. But another way (Method Ⅱ in the textbook) 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, of course.)

Similarly, there are two ways to solve an equation between two fractions (a rational equation, assuming that the top and bottom of both sides are polynomials, so that each side is a rational expression). The more direct way is to cross-multiply: simplify each side separately, then multiply the top of each side by the bottom of the other side, setting 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.) This corresponds to Method Ⅰ for simplifying complex fractions.

Then there is another method of solving rational equations, which corresponds to Method Ⅱ for simplifying complex fractions. Now you factor only 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, of course.) This is covered in Section 7.7 of the textbook.

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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