Rational Expressions Worksheet Answer Key – Algebra 2

Problem 1 Simplifying monomial fractions

Divide the numbers by 6 and cancel powers: 12/18 = 2/3, x³/x = x², y/y² = 1/y.

2x²3y

Problem 2 Excluded values

a) x² − 9 = (x − 3)(x + 3), so x ≠ 3 and x ≠ −3.

b) 2x − 7 = 0 at x = 72.

c) x² + x − 12 = (x + 4)(x − 3), so x ≠ −4 and x ≠ 3.

(a) ±3; (b) 72; (c) −4 and 3

Problem 3 Simplifying by factoring

Numerator: (x − 5)(x + 5). Denominator: (x − 5)(x + 2).

Cancel x − 5: x + 5x + 2, with x ≠ 5 and x ≠ −2.

Problem 4 Multiplying rational expressions

The factor x + 5 cancels, and 3x²/(9x) = x/3.

x3, with x ≠ −5 and x ≠ 0.

Problem 5 Excluded values on a number line

x² − x − 20 = (x − 5)(x + 4) = 0 at x = 5 and x = −4.

Domain: all real numbers except −4 and 5

Problem 6 Dividing rational expressions

Multiply by the reciprocal: (x − 2)(x + 2)x + 3 · x(x + 3)x + 2.

Cancel x + 2 and x + 3 to get x(x − 2), with x ≠ −3, −2, 0.

x(x − 2)

Problem 7 Like denominators

a) 3x + 1 + 2x − 9x − 4 = 5x − 8x − 4.

b) Distribute the minus: 7x + 2 − x + 10x + 6 = 6x + 12x + 6 = 6(x + 2)x + 6.

Problem 8 Unlike denominators

The LCD is (x + 5)(x − 3).

7(x − 3) + 2(x + 5)(x + 5)(x − 3) = 7x − 21 + 2x + 10(x + 5)(x − 3).

9x − 11(x + 5)(x − 3)

Problem 9 Factoring a denominator first

x² − 9 = (x − 3)(x + 3), so the LCD is (x − 3)(x + 3).

5 − 2(x + 3)(x − 3)(x + 3) = 5 − 2x − 6(x − 3)(x + 3).

−2x − 1(x − 3)(x + 3)

Problem 10 Adding with a quadratic denominator

x² − 5x + 6 = (x − 2)(x − 3), so the second fraction needs the factor x − 3.

x + 4(x − 3)(x − 2)(x − 3) = x + 4x − 12(x − 2)(x − 3).

5x − 12(x − 2)(x − 3)

Problem 11 A first rational equation

Multiply every term by x: 8 − 3x = 2, so 3x = 6 and x = 2.

Check: 8/2 − 3 = 1 and 2/2 = 1 ✓.

x = 2

Problem 12 No solution

Multiply by x − 5: x + 2(x − 5) = 5, so 3x = 15 and x = 5.

But x = 5 makes the denominator 0, so it is extraneous.

No solution

Problem 13 Solving with a graph

The graphs cross at (3, 2) and (−2, −3).

Algebra: multiply by x: 6 = x² − x, so x² − x − 6 = 0 and (x − 3)(x + 2) = 0.

x = 3 and x = −2

Problem 14 Two denominators and a quadratic

Multiply by (x − 1)(x + 4): 3(x + 4) + 2(x − 1) = (x − 1)(x + 4).

5x + 10 = x² + 3x − 4, so x² − 2x − 14 = 0.

x = 2 ± √(4 + 56)2 = 2 ± 2√152 = 1 ± √15.

Neither value equals 1 or −4, so both are valid.

x = 1 + √15 or x = 1 − √15

Problem 15 Rejecting an extraneous solution

The LCD is (x − 2)(x + 2). Multiply: 3(x + 2) − 12 = x² − 4, so 3x − 6 = x² − 4 and x² − 3x + 2 = 0.

(x − 1)(x − 2) = 0, so x = 1 or x = 2.

x = 2 makes a denominator 0, so reject it. Check x = 1: −3 + 4 = 1 ✓.

x = 1

Problem 16 Painting a fence together

Ana: 16 per hour. Ben: 19 per hour. Together: 16 + 19 = 318 + 218 = 518 per hour.

Time = 1 ÷ 518 = 185 = 3.6 hours.

3.6 hours (3 hours 36 minutes)

Problem 17 Two hoses filling a pond

Let t be the hours for the faster hose; the slower takes t + 5. Rates: 1t + 1t + 5 = 16.

Multiply by 6t(t + 5): 6(t + 5) + 6t = t(t + 5), so 12t + 30 = t² + 5t.

t² − 7t − 30 = 0, which factors as (t − 10)(t + 3) = 0. The time must be positive, so t = 10.

Check: 1/10 + 1/15 = 5/30 = 1/6 ✓.

Faster hose 10 hours; slower hose 15 hours

Problem 18 Paddling against a current

Let s be the still-water speed. Upstream speed s − 3, downstream s + 3. Total time:

12s − 3 + 12s + 3 = 3.

Multiply by (s − 3)(s + 3): 12(s + 3) + 12(s − 3) = 3(s² − 9), so 24s = 3s² − 27 and s² − 8s − 9 = 0.

(s − 9)(s + 1) = 0. A speed must be positive, so s = 9.

Check: 12/6 + 12/12 = 2 + 1 = 3 ✓.

9 mph

Problem 19 Resistors in parallel

1x + 1x + 6 = 14. Multiply by 4x(x + 6): 4(x + 6) + 4x = x(x + 6).

8x + 24 = x² + 6x, so x² − 2x − 24 = 0 and (x − 6)(x + 4) = 0. A resistance is positive, so x = 6.

Check: 1/6 + 1/12 = 3/12 = 1/4 ✓.

6 ohms and 12 ohms

Problem 20 A multi-step quotient

Rewrite as a product: (x − 3)(x + 3)x(x + 2) · (x − 2)(x + 2)x + 3.

Cancel x + 3 and x + 2: (x − 3)(x − 2)x.

Excluded: x = 0 and x = −2 (original denominator), x = −3 (divisor is 0), and x = ±2 (from x² − 4).

(x − 3)(x − 2)x, with x ≠ 0, −2, −3, 2