ACT Preparation / Unit 8 ACT 0720 worked example, 60 to 90 seconds
Adding Two Rational Expressions
The idea
To add or subtract two rational expressions, work in four steps. First, factor each denominator. Second, build the least common denominator from every factor, each to its highest power. Third, multiply each numerator by whatever factors its denominator lacks. Fourth, combine the numerators over the common denominator, simplify, and factor again to see whether anything cancels.
In practice
Worked example: 3/(x² − 4) − 1/(x − 2). Step one: x² − 4 = (x + 2)(x − 2). Step two: the LCD is (x + 2)(x − 2). Step three: the second fraction needs x + 2, becoming (x + 2)/((x + 2)(x − 2)). Step four: (3 − (x + 2))/((x + 2)(x − 2)) = (1 − x)/((x + 2)(x − 2)).
Where people slip
Subtracting a numerator subtracts every term of it. Writing 3 − x + 2 instead of 3 − x − 2 is the slip to watch for, and it gives 5 − x instead of 1 − x.
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