WRITE AS A SINGLE FRACTION

Example 1 :

2/x + 2 – 4/2x + 1

Solution :

2/(x + 2) – 4/(2x + 1)

Since denominators are not same, we take the LCM of  (x + 2) and (2x + 1)

LCM = (x + 2)(2x + 1)

= 2(2x + 1) - 4(x + 2) / (x + 2)(2x + 1)

Using distributive property, we get

= (4x + 2 - 4x - 8) / (x + 2)(2x + 1)

= -6/ (x + 2)(2x + 1)

Example 2 :

5/x – 2/x+3

Solution :

5/x – 2/x+3

Since denominators are not same, we take the LCM of  (x) and (x+3)

LCM = x(x + 3)

= [5(x + 3) - 2x] / x (x + 3)

Using distributive property, we get

= (5x + 15 - 2x)/x(x + 3)

= (3x + 15) / x(x + 3)

Example 3 :

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

Solution :

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

Since denominators are not same. we take the LCM of  (x+2) and (x-4)

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

= [x(x - 4) + 3(x + 2)] / (x + 2)(x - 4)

= (x² - 4x + 3x + 6) / (x + 2)(x - 4)

= (x² - x + 6) / (x + 2)(x - 4)

Example 4 :

2 + 4/x-3

Solution :

2/1 + 4/x-3

Since denominators are not same. we take LCM of 1 and (x-3)

LCM = (x - 3)

= [2(x - 3) + 4]/(x - 3)

= (2x - 6 + 4) / (x - 3)

= (2x - 2) / (x - 3)

= 2(x - 1) / (x - 3)

Example 5 :

[3x/x+2] - 1

Solution :

[3x/x+2] - 1

Since denominators are not same, we take the LCM of 1 and (x+2)

LCM = x + 2

= [3x - 1(x + 2)] / (x + 2)

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

= (2x - 2) / (x + 2)

= 2(x - 1) / (x + 2)

Example 6 :

x/(x+3) + x-1/(x+2)

Solution :

[x/(x+3)] + [x-1/(x+2)]

Since denominators are not same, we take the LCM of (x + 3) and (x + 2)

LCM = (x + 3)(x + 2)

= [x(x + 2) + (x - 1)(x + 3)]/(x + 3)(x + 2)]

= (x² + 2x + x² + 3x - x - 3) / (x + 3)(x + 2)

= (2x² + 4x - 3) / (x + 3)(x + 2)

Example 7 :

2/x(x+1) + 1/(x+1)

Solution :

2/x(x+1) + 1/(x+1)

Since denominators are not same, we take LCM of x(x+1) and x

LCM = x(x + 1)

= (2 + 1x)/x (x + 1)

= (2 + x) / x(x + 1)

Example 8 :

1/(x-1) - 1/x + 1/(x+1)

Solution :

1/(x-1) - 1/x + 1/(x+1)

Since denominators are not same, we take LCM of x-1, x and x+1

LCM = x(x - 1)(x + 1)

= [1x(x + 1) - 1(x + 1)(x - 1) + x(x - 1)] / x(x - 1)(x + 1)

= [x² + x - (x² - 1) + x² - x] / x(x - 1)(x + 1)

= [x² + x - (x² - 1) + x² - x] / x(x - 1)(x + 1)

= [x² + x - x² + 1 + x² - x] / x(x - 1)(x + 1)

= (x²+ 1) / x(x - 1)(x + 1)

= (x² + 1)/x(x - 1)(x + 1)

Example 9 :

x/(x-1) - 1/x + x/(x+1)

Solution :

x/(x-1) - 1/x + x/(x+1)

Since denominators are not same, we take the LCM of x-1, x and x+1

LCM = x(x + 1)(x - 1)

= x²(x + 1) - 1(x + 1)(x - 1) + x²(x - 1)/x(x-1)(x + 1)

= [x³ + x² - 1(x + 1)(x - 1) + x³ - x²)] / x(x-1)(x + 1)

= [x³ + x² - x² + 1 + x³ - x²)] / x(x-1)(x + 1)

= [2x³ - x² + 1] / x(x-1)(x + 1)

Example 10 :

2/x+1 - 1/x-1 + 3/x+2

Solution :

2/x+1 - 1/x-1 + 3/x+2

Since denominators are not same, we take the LCM of x+1, x-1 and x+2

LCM = (x + 1)(x - 1)(x + 2)

= [2(x - 1)(x + 2) - 1(x + 1)(x + 2) + 3(x + 1) (x - 1)]/(x + 1)(x - 1)(x + 2)

Simplifying the numerator,

= 2(x² + x - 2) - 1(x² + 3x + 2) + 3(x² - 1)

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

= 4x² - x - 9

Writing it as fraction, we get

= (4x² - x - 9) / (x + 1)(x - 1)(x + 2)

Example 11 :

16/(w - 5) - [18/w] = 40/w

Solution :

16/(w - 5) - [18/w] = 40/w

16/(w - 5) = 40/w + (18/w)

16/(w - 5) = 58/w

Doing cross multiplication, we get

16w = 58 (w - 5)

16w = 58w - 290

16w - 58w = -290

-42w = -290

w = 290/42

w = 145/21

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