SOLVING RATIONAL EQUATIONS WITH FACTORING WORKSHEET

Problem 1 :

3x + xx + 2 = 4x2 + 2x

Problem 2 :

xx + 1 + 5x = 1x2 + x

Problem 3 :

5x + 2 - 3x - 2 = 2x4 - x2

Problem 4 :

y + 3y + 2 - yy2 - 4 = yy - 2

Problem 5 :

2a + 4 + 2a - 1a2 + 2a - 8 = 1a - 2

Problem 6 :

3x2 - 6x + 9 + x - 23x - 9 = x2x - 6

Problem 7 :

3x2 - x - 12 + 2x2 + x - 6 = 4x2 - 6x + 8

Problem 8 :

2x + 3 - 3x + 5x2 + 4x + 3 = 5x + 1

Problem 9 :

6x2 - 2x - 3 - 1x2 - 1 = 2x2 - 4x + 3

Solutions

1. Answer

Given, 3x + xx + 2 = 4x2 + 2x 3x × x + 2x + 2 + x x + 2 × xx= 4x2 + 2x3(x + 2)x(x + 2) + x2x(x + 2) = 4x(x + 2)3(x + 2) + x2x(x + 2) = 4x(x + 2)3x + 6 + x2x(x + 2) = 4x(x + 2)3x + 6 + x2x(x + 2) = 4x(x + 2) Multiplying x(x + 2) on each sides.3x + 6 + x2x(x + 2) × x(x + 2) = 4x(x + 2)× x(x + 2)

3x + 6 + x2 = 4

3x + 6 + x2 - 4 = 0

x2 + 3x + 2 = 0

By using algebraic expression.

x2 + x + 2x  + 2 = 0

x(x + 1)  + 2(x + 1) = 0

x + 1 = 0 and x + 2 = 0

x = -1, x = -2

2. Answer

Given, xx + 1 + 5x = 1x2 + xxx + 1 × xx + 5x × x + 1x + 1= 1x2 + xx2x(x + 1) + 5(x + 1)x(x + 1) = 1x2 + xx2 + 5(x + 1)x(x + 1) = 1x2 + xx2 + 5x + 5x2 + x = 1x2 + xMultiplying x2 + x on each sides.x2 + 5x + 5x2 + x × x2 + x = 1x2 + x × x2 + x

x2 + 5x + 5 = 1

x2 + 5x + 5 - 1 = 0

x2 + 5x + 4 = 0

By using algebraic expression.

x2 + x + 4x + 4 = 0

x(x + 1) + 4(x + 1) = 0

x + 4 = 0 and x + 1 = 0

x = -4, x = -1

3. Answer

Given, 5x + 2 - 3x - 2 = 2x4 - x25x + 2 × x - 2x - 2 - 3x - 2 × x + 2x + 2 = 2x22 - x25(x - 2) - 3(x + 2)(x + 2) (x - 2) = 2x(2 - x) (2 + x)5x - 10 - 3x - 6(x + 2) (x - 2) = 2x[-(x - 2)] (x + 2)Multiplying (x + 2) (x - 2) on each sides.5x - 10 - 3x - 6(x + 2) (x - 2) × (x + 2) (x - 2) = 2x[-(x - 2)] (x + 2) × (x + 2) (x - 2) 5x - 10 - 3x - 6 = 2x-1

5x - 10 - 3x - 6 = -2x

2x - 16 = -2x

2x - 16 + 2x = 0

4x - 16 = 0

4x = 16

x = 16/4

x = 4

4. Answer

Given, y + 3y + 2 - yy2 - 4 = yy - 2y + 3y + 2 - yy2 - 22 = yy - 2y + 3y + 2 - y(y + 2)(y - 2) = yy - 2y + 3y + 2 × y - 2y - 2 - y(y + 2)(y - 2) = yy - 2(y + 3)(y - 2)(y + 2)(y - 2) - y(y + 2)(y - 2) = yy - 2(y + 3)(y - 2) - y(y + 2)(y - 2) = yy - 2Multiplying y - 2 on each sides.(y + 3)(y - 2) - y(y + 2)(y - 2) × y - 2= yy - 2 × y -2(y + 3)(y - 2) - yy + 2 = y

By using cross - multiplication.We get

(y + 3)(y - 2) - y = y(y + 2)

y2 - 2y + 3y - 6 - y = y2 + 2y

y2 + y - 6 - y = y2 + 2y

y2 - 6 = y2 + 2y 

y2 - 6 - y2 - 2y = 0

-2y - 6 = 0

-2y = 6

y = -6/2

y = - 3

5. Answer

Given, 2a + 4 + 2a - 1a2 + 2a - 8 = 1a - 22a + 4 + 2a - 1(a + 4)(a - 2) = 1a - 22a + 4 × a - 2a - 2+ 2a - 1(a + 4)(a - 2) = 1a - 22(a - 2)(a + 4)(a - 2) + 2a - 1(a + 4)(a - 2) = 1a - 22a - 4 + 2a - 1(a + 4)(a - 2) = 1a - 2Multiplying a - 2 on each sides.2a - 4 + 2a - 1(a + 4)(a - 2) × a - 2 = 1a - 2 × a - 22a - 4 + 2a - 1a + 4 = 1

By using cross - multiplication.We get

2a - 4 + 2a - 1 = a + 4

2a - 4 + 2a - 1 - a - 4 = 0

3a - 9 = 0

3a = 9

a = 9/3

a = 3

6. Answer

Given, 3x2 - 6x + 9 + x - 23x - 9 = x2x - 63(x - 3)2 + x - 23(x - 3) = x2(x - 3)3(x - 3)2 × 33 + x - 23(x - 3) × x - 3x - 3 = x2(x - 3)93(x - 3)2 + (x - 2)(x - 3)3(x - 3)(x - 3) = x2(x - 3)93(x - 3)2 + (x - 2)(x - 3)3(x - 3)2 = x2(x - 3) 9 + (x - 2)(x - 3)3(x - 3)2 = x2(x - 3)9 + x2 - 3x - 2x + 6 3(x - 3)2 = x2(x - 3)x2 - 5x + 15 3(x - 3)2 = x2(x - 3)Multiplying 6(x - 3)2 on each sides.x2 - 5x + 15 3(x - 3)2 × 6(x - 3)2= x2(x - 3) × 6(x - 3)2

2(x2 - 5x + 15) = 3x(x  - 3)

2x2 - 10x + 30 = 3x2 - 9

2x2 - 10x + 30 - 3x2 + 9 = 0

-x2 - x + 30 = 0

-(x2 + x - 30) = 0

x2 + x - 30 = 0

By using algebraic expression.

x2 + 6x + 5x - 30 = 0

x(x + 6) - 5(x + 6) = 0

(x - 5) and (x + 6) = 0

x = 5, x = -6

7. Answer

Given, 3x2 - x - 12 + 2x2 + x - 6 = 4x2 - 6x + 8 3(x - 4)(x + 3) + 2(x + 3)(x - 2) = 4(x - 4)(x - 2)3(x - 4)(x + 3) × x - 2x - 2 + 2(x + 3)(x - 2) × x - 4x - 4 = 4(x - 4)(x - 2)3(x - 2)(x - 4)(x + 3) + 2(x - 4)(x + 3)(x - 2) = 4(x - 4)(x - 2)3(x - 2) + 2(x - 4)(x - 4)(x + 3)(x - 2) = 4(x - 4)(x - 2)Multiplying (x - 4) (x - 2) on each sides.3(x - 2) + 2(x - 4)(x - 4)(x + 3)(x - 2) × (x - 4)(x - 2)= 4(x - 4)(x - 2) × (x - 4)(x - 2)3(x - 2) + 2(x - 4)x + 3 = 4

By using cross - multiplication.We get

3(x - 2) + 2(x - 4) = 4(x + 3)

3x - 6 + 2x - 8 = 4x + 12

3x - 6 + 2x - 8 - 4x - 12 = 0

x - 14 - 12 = 0

x - 26 = 0

x = 26

8. Answer

Given, 2x + 3 - 3x + 5x2 + 4x + 3 = 5x + 1 2x + 3 - 3x + 5(x + 3)(x + 1) = 5x + 12(x + 3) × x + 1x + 1- 3x + 5(x + 3)(x + 1) = 5x + 1 2(x + 1)(x + 3)(x + 1) - 3x + 5(x + 3)(x + 1) = 5x + 1 2(x + 1) -(3x + 5)(x + 3)(x + 1) = 5x + 1 Multiplying x + 1 on each sides.2(x + 1) - (3x + 5)(x + 3)(x + 1) × x + 1 = 5x + 1 × x + 12(x + 1) - (3x + 5)x + 3 = 5

By using cross - multiplication.We get

2(x + 1) - (3x + 5) = 5(x + 3)

2x + 2 - 3x - 5 = 5x + 15

2x + 2 - 3x - 5 - 5x - 15 = 0

-6x - 18 = 0

-6x = 18

x = -18/6

x = -3

9. Answer

Given, 6x2 - 2x - 3 - 1x2 - 1 = 2x2 - 4x + 36(x + 1)(x - 3) - 1x2 - 12 = 2(x - 1)(x - 3)6(x + 1)(x - 3) - 1(x - 1)(x + 1) = 2(x - 1)(x - 3)6(x + 1)(x - 3) × x - 1x - 1- 1(x - 1)(x + 1) × x - 3x - 3= 2(x - 1)(x - 3) 6(x - 1) - (x - 3)(x - 1)(x + 1)(x - 3) = 2(x - 1)(x - 3)Multiplying (x - 1)(x - 3) on each sides.6(x - 1) - (x - 3)(x - 1)(x + 1)(x - 3) × (x - 1)(x - 3)= 2(x - 1)(x - 3) × (x - 1)(x - 3)6(x - 1) - (x - 3)(x + 1) = 2

By using cross - multiplication.We get

6(x - 1) - (x  - 3) = 2(x + 1)

6x - 6 - x + 3 = 2x  + 2

6x - 6 - x + 3 - 2x - 2 = 0

3x - 5 = 0

3x = 5

x = 5/3

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