Lesson 5 · from §64 of the 1910 text
Adding Fractions: Watch the Signs
Sometimes two denominators are negatives of each other, like x - 3 and 3 - x. Don't multiply them together! Use the sign principle to flip one.
Example. \dfrac{x - 1}{2 - x} + \dfrac{x^2 - 3}{x^2 - 4}. The denominators look like (2 - x) and (x + 2)(x - 2). Multiply both terms of the first fraction by -1: \dfrac{1 - x}{x - 2}. Now the L.C.D. is just (x + 2)(x - 2): \frac{(1 - x)(x + 2) + (x^2 - 3)}{(x + 2)(x - 2)} = \frac{-x^2 - x + 2 + x^2 - 3}{(x + 2)(x - 2)} = \frac{-x - 1}{x^2 - 4}, \text{ or } \frac{x + 1}{4 - x^2}.
Signs in products
Changing the sign of one factor of a product changes the sign of the whole product: (x - 2)(x - 3) = x^2 - 5x + 6, but (2 - x)(x - 3) = -x^2 + 5x - 6.
Changing the signs of two factors leaves the product alone: (2 - x)(3 - x) = x^2 - 5x + 6.
An odd number of sign changes among the factors flips the sign of the product; an even number leaves it unchanged.
Example. In \dfrac{1}{(a - b)(a - c)} + \dfrac{1}{(b - c)(b - a)} - \dfrac{1}{(c - a)(c - b)}, flip one factor in the second denominator (b - a \to a - b, which flips the fraction's sign) and both factors in the third (no change). The fractions become \frac{1}{(a - b)(a - c)} - \frac{1}{(b - c)(a - b)} - \frac{1}{(a - c)(b - c)}, with L.C.D. (a - b)(a - c)(b - c).
Quick check
\displaystyle \frac{5}{x-2}+\frac{3}{2-x}
Exercises
How to type math
Use ^ for powers (a^2), / for fractions ((a+b)/2), sqrt(x) for square roots, and cbrt(x) for cube roots. You can leave out the multiplication sign: 3ab means 3 \cdot a \cdot b. Mixed numbers like 4 1/2 work too. The preview under each box shows how your answer was read.
Find the algebraic sum. Watch for denominators that are negatives of each other.
\displaystyle \frac{x-1}{2-x}+\frac{x^2-3}{x^2-4}
\displaystyle \frac{5}{x-3}+\frac{3}{3-x}
\displaystyle \frac{6}{x^2-25}-\frac{3}{5-x}
\displaystyle \frac{3a}{a^2-4}+\frac{2a-1}{2-a}
\displaystyle \frac{3c}{9-c^2}-\frac{4c-2}{c-3}
\displaystyle \frac{x}{2x-1}-\frac{x}{1+2x}-\frac{x}{1-4x^2}
\displaystyle \frac{7}{x^2-13x+42}+\frac{4x-1}{7-x}-\frac{2x+3}{6-x}
\displaystyle \frac{3x-1}{x^2+7x-8}-\frac{4x-1}{1-x}+\frac{x+2}{8+x}
Products of factors in the denominators.
\displaystyle \frac{1}{(a-b)(a-c)}+\frac{1}{(b-c)(b-a)}-\frac{1}{(c-a)(c-b)}
\displaystyle \frac{2}{(x-y)(x-z)}-\frac{1}{(x-y)(z-x)}
\displaystyle \frac{3a}{(a-3)(a-4)}-\frac{2a}{(3-a)(4-a)}
\displaystyle \frac{2}{(m-n)(m+n)}+\frac{3}{(n-m)(m-7)}-\frac{4}{(n-m)(7-m)}
\displaystyle \frac{c}{c^2-10c+24}-\frac{1}{6c-c^2-8}+\frac{3}{(6-c)(2-c)}