SIMPLIFYING COMPLEX FRACTIONS WITH VARIABLES

Simplify the fractions in the numerator and the denominator separately, then divide the two fractions and simplify.

Simplify the following :

Example 1 :

(1 + 1/x) /(1 - 1/x²)

Solution :

(1) / (2)

= [(x + 1)/x] / [(x + 1) (x - 1)/x²]

= [(x + 1)/x] ⋅ [x²/(x + 1) (x - 1)]

= x/(x - 1)

Example 2 :

[4 + 12/(2x - 3)] / [5 + 15/(2x - 3)]

Solution :

Simplifying the numerator :

= [4 + 12/(2x - 3)]

= [4(2x - 3) + 12] / (2x - 3)

= (8x - 12 + 12) / (2x - 3)

= 8x / (2x - 3)  -------(1)

Simplifying the denominator :

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

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

= (10x - 15 + 15) / (2x - 3)

= 10x / (2x - 3)  -------(2)

(1) / (2)

= 8x / (2x - 3) / 10x / (2x - 3)

= [8x / (2x - 3)]  ⋅ [(2x - 3)/10x]

= 4/5

Example 3 :

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

Solution :

Simplifying the numerator :

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

Least common multiple of x and (x + 3) is x (x + 3)

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

= (2x + 6 - 5x) / x(x + 3)

= (-3x + 6) / x(x + 3) ------(1)

Simplifying the denominator :

= [(3/x) + 3/(x+3)]

= [3(x + 3) + 3x] / x(x + 3)

= (3x + 9 + 3x) / x(x + 3)

= (6x + 9) / x(x + 3)

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

(1) / (2)

= [(-3x + 6) / x(x + 3)] / [3(2x + 3) / x(x + 3)]

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

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

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

Example 4 :

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

Solution :

Simplifying the numerator :

= [x/(x + 2) - x/(x - 2)]

Taking the least common multiple, we get

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

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

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

Simplifying the denominator :

=  [x/(x + 2) + x/(x - 2)]

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

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

= 2x² / (x + 2) (x - 2) -----(2)

(1) / (2)

= -4x / 2x²

= -2/x