Operations on rational algebraic fractions
Answers are for parents and teachers.
1
Add 4 ÷ (x + 2) + 1 ÷ (x − 2) and give the domain.
(4(x − 2) + (x + 2)) ÷ (x² − 4) = (5x − 6) ÷ (x² − 4). Domain R − {2, −2}.
2
Multiply (x² − 9) ÷ (x² − 4) × (x + 2) ÷ (x + 3) and give the domain.
(x − 3)(x + 3)(x + 2) ÷ ((x − 2)(x + 2)(x + 3)) = (x − 3) ÷ (x − 2). Domain R − {2, −2, −3}.
3
Divide (x + 2) ÷ x by (x² − 4) ÷ x² and give the domain.
(x + 2) ÷ x × x² ÷ ((x − 2)(x + 2)) = x ÷ (x − 2). Domain: x = 0 (denominators) and x = ±2 (divisor’s numerator): R − {0, 2, −2}.
4
Find the additive inverse and the multiplicative inverse of (3 − x) ÷ (x + 4), with the domain of the second.
Additive: (x − 3) ÷ (x + 4). Multiplicative: (x + 4) ÷ (3 − x), domain R − {3, −4}.
5
Let f(x) = (x + 3) ÷ (x − 2). Solve 1 ÷ f(x) = 1/2.
(x − 2) ÷ (x + 3) = 1/2, so 2x − 4 = x + 3 and x = 7. Domain R − {−3, 2}. Check: 5 ÷ 10 = 1/2 ✓.