Rational equations
Answers are for parents and teachers.
1
Solve 4/(x − 3) = 5/(x + 1).
4(x + 1) = 5(x − 3) ⇒ x = 19; no denominator is zero. Answer: 19.
2
Solve (x² − 3x − 10)/(x − 5) = 0.
x² − 3x − 10 = (x − 5)(x + 2); roots 5, −2; x = 5 is extraneous. Answer: x = −2.
3
Solve 1/x + 1/(x + 1) = 3/2.
2(x + 1) + 2x = 3x(x + 1) ⇒ 3x² − x − 2 = 0, D = 25, x = 1 or x = −2/3; neither makes a denominator zero.
4
Solve (x/(x − 1))² − 5 · x/(x − 1) + 6 = 0 by substitution.
t = x/(x − 1): t² − 5t + 6 = 0, t = 2 or 3. t = 2 ⇒ x = 2; t = 3 ⇒ x = 3/2. Answer: 2; 3/2.
5
Solve the system: 1/x + 1/y = 5/6, xy = 6.
(x + y)/(xy) = 5/6 ⇒ x + y = 5; xy = 6 ⇒ (2, 3) and (3, 2).