Rational expressions

Simplifying, operating on and solving with algebraic fractions.

5 sheets · 40 questions · one answer key · 7 pages

Recommended for grades 10–11

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$Rational expressions pack — Solving: one fraction worksheet$

Rational expressions Sheet 1 of 5

Rational expressionsRoots to 20

Name: Date:

Factor the top and the bottom, then cancel.

1x² − 18x + 72x² − 4x − 12

2x² − 5x + 4x² − 11x + 10

3x² − 29x + 208x² − 21x + 104

4x² − 15x + 36x² + 10x − 39

5x² + 25x + 154x² + 19x + 88

6x² + 5x − 234x² − 31x + 234

7x² + x − 2x² + 9x − 10

8x² + 2x − 323x² − 16x − 17

Rational × ÷Multiplying

Name: Date:

Multiply and cancel. Write each answer as one fraction.

1(x − 9)(x − 6)(x − 6)(x + 2)

2(x + 8)(x + 9)(x + 9)(x + 5)

3(x + 9)(x − 1)(x − 1)(x − 6)

4(x − 6)(x − 8)(x − 8)(x + 9)

5(x + 7)(x + 4)(x + 4)(x + 3)

6(x + 7)(x + 1)(x + 1)(x − 8)

7(x − 7)(x + 9)(x + 9)(x + 1)

8(x − 3)(x − 1)(x − 1)(x + 5)

Rational × ÷Dividing

Name: Date:

Divide and cancel. Write each answer as one fraction.

1(x + 8)(x − 6)(x − 8)(x − 6)

2(x + 8)x(x − 7)x

3(x + 3)(x − 8)(x + 6)(x − 8)

4(x − 6)(x + 4)(x + 9)(x + 4)

5(x + 2)(x − 3)(x − 2)(x − 3)

6(x − 8)(x + 4)x(x + 4)

7(x + 8)(x − 9)(x + 4)(x − 9)

8(x − 6)(x + 8)(x + 2)(x + 8)

Rational equationsOne fraction

Name: Date:

Solve each equation for x.

121(x − 1)7x =

2−60(x − 3)6x =

3−18(x − 2)9x =

4−49(x − 3)7x =

5−16(x + 7)8x =

632(x + 1)8x =

78(x + 5)8x =

8−63(x − 9)9x =

Rational equationsA fraction on each side

Name: Date:

Cross-multiply and solve each equation for x.

13(x − 3)−18(x + 4)x =

22(x − 2)−7(x + 7)x =

313(x − 7)10(x − 4)x =

46(x − 5)−8(x + 2)x =

520(x + 2)44(x + 8)x =

65(x − 9)−30(x − 2)x =

76(x + 8)−60(x − 3)x =

87(x + 6)−14(x − 3)x =

Power Pack · Rational expressionsAnswer key

Rational expressions

1. (x² − 18x + 72) / (x² − 4x − 12) = (x − 12) / (x + 2)2. (x² − 5x + 4) / (x² − 11x + 10) = (x − 4) / (x − 10)3. (x² − 29x + 208) / (x² − 21x + 104) = (x − 16) / (x − 8)4. (x² − 15x + 36) / (x² + 10x − 39) = (x − 12) / (x + 13)5. (x² + 25x + 154) / (x² + 19x + 88) = (x + 14) / (x + 8)6. (x² + 5x − 234) / (x² − 31x + 234) = (x + 18) / (x − 18)7. (x² + x − 2) / (x² + 9x − 10) = (x + 2) / (x + 10)8. (x² + 2x − 323) / (x² − 16x − 17) = (x + 19) / (x + 1)

Multiplying

1. (x − 9)/(x − 6) × (x − 6)/(x + 2) = (x − 9) / (x + 2) (x ≠ −2, 6)2. (x + 8)/(x + 9) × (x + 9)/(x + 5) = (x + 8) / (x + 5) (x ≠ −9, −5)3. (x + 9)/(x − 1) × (x − 1)/(x − 6) = (x + 9) / (x − 6) (x ≠ 1, 6)4. (x − 6)/(x − 8) × (x − 8)/(x + 9) = (x − 6) / (x + 9) (x ≠ −9, 8)5. (x + 7)/(x + 4) × (x + 4)/(x + 3) = (x + 7) / (x + 3) (x ≠ −4, −3)6. (x + 7)/(x + 1) × (x + 1)/(x − 8) = (x + 7) / (x − 8) (x ≠ −1, 8)7. (x − 7)/(x + 9) × (x + 9)/(x + 1) = (x − 7) / (x + 1) (x ≠ −9, −1)8. (x − 3)/(x − 1) × (x − 1)/(x + 5) = (x − 3) / (x + 5) (x ≠ −5, 1)

Dividing

1. (x + 8)/(x − 6) ÷ (x − 8)/(x − 6) = (x + 8) / (x − 8) (x ≠ 6, 8)2. (x + 8)/x ÷ (x − 7)/x = (x + 8) / (x − 7) (x ≠ 0, 7)3. (x + 3)/(x − 8) ÷ (x + 6)/(x − 8) = (x + 3) / (x + 6) (x ≠ −6, 8)4. (x − 6)/(x + 4) ÷ (x + 9)/(x + 4) = (x − 6) / (x + 9) (x ≠ −9, −4)5. (x + 2)/(x − 3) ÷ (x − 2)/(x − 3) = (x + 2) / (x − 2) (x ≠ 2, 3)6. (x − 8)/(x + 4) ÷ x/(x + 4) = (x − 8) / x (x ≠ −4, 0)7. (x + 8)/(x − 9) ÷ (x + 4)/(x − 9) = (x + 8) / (x + 4) (x ≠ −4, 9)8. (x − 6)/(x + 8) ÷ (x + 2)/(x + 8) = (x − 6) / (x + 2) (x ≠ −8, −2)

Solving: one fraction

1. 21/(x − 1) = 7 → x = 4 (x ≠ 1)2. −60/(x − 3) = 6 → x = −7 (x ≠ 3)3. −18/(x − 2) = 9 → x = 0 (x ≠ 2)4. −49/(x − 3) = 7 → x = −4 (x ≠ 3)5. −16/(x + 7) = 8 → x = −9 (x ≠ −7)6. 32/(x + 1) = 8 → x = 3 (x ≠ −1)7. 8/(x + 5) = 8 → x = −4 (x ≠ −5)8. −63/(x − 9) = 9 → x = 2 (x ≠ 9)

Solving: cross-multiply

1. 3/(x − 3) = −18/(x + 4) → x = 2 (x ≠ −4, 3)2. 2/(x − 2) = −7/(x + 7) → x = 0 (x ≠ −7, 2)3. 13/(x − 7) = 10/(x − 4) → x = −6 (x ≠ 4, 7)4. 6/(x − 5) = −8/(x + 2) → x = 2 (x ≠ −2, 5)

Power Pack · Rational expressionsAnswer key

Solving: cross-multiply

5. 20/(x + 2) = 44/(x + 8) → x = 3 (x ≠ −8, −2)6. 5/(x − 9) = −30/(x − 2) → x = 8 (x ≠ 2, 9)7. 6/(x + 8) = −60/(x − 3) → x = −7 (x ≠ −8, 3)8. 7/(x + 6) = −14/(x − 3) → x = −3 (x ≠ −6, 3)

Rational expressions

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