1. (5x + 7)/(2x + 1) = (x + 1)/(3x + 5)

Difference between numerator and denominator 

= 5x + 7 – 2x – 1 

= 3x + 6 = 3(x + 2) …(i) 

Difference between numerator and denominator is

R.H.S. = 3x + 5 – x – 1 

= 2x + 4 = 2(x + 2) …(ii) 

Ratio of (i) and (ii) is 3 : 2 

Thus, by formula 3(x + 2) = 0 

⇒ x + 2 = 0 

⇒ x = -2

2. (3x + 6)/(6x + 3) = (5x + 4)/(2x + 7)

Difference between numerator and denominator in L.H.S. 

= 6x + 3 – 3x – 6 = 3x – 3 …(i) 

Difference numerator and denominator is L.H.S. 

= 5x + 4 – 2x – 7 = 3x – 3 …(ii) 

Difference numerator and denominator in R.H.S. 3x – 3 = 0 

⇒ 3x = 3 

⇒ x = 1 

Thus, x = 1

(i) and (ii) are equal, so by formula 

3x + 6 + 5x + 4 = 8x + 10 …(i) 

Adding denominators of both sides 

6x + 3 + 2x + 7 = 8x + 10 …(ii) 

(i) and (ii) are equal so by formula. 

8x + 10 = 0 

⇒ 8x = – 10 

⇒ x = -10/8 = -5/4

thus, x = -5/4 and x = 1

3. 1/(x + 2) + 1/(x + 6) = 1/(x + 1) + 1/(x + 7)

Sum of denominators of L.H.S. 

= x + 2 + x + 6 

= 2x + 8 …(i) 

Sum of denominators of R.H.S. 

= x + 1 + x + 7 

= 2x + 8 …(ii) 

(i) and (ii) are equal so by formula 

2x + 8 = 0 

⇒ 2x = -8 

⇒ x = -4

4. 1/(x - 4) + 1/(x - 6) = 1/(x - 2) + 1/(x - 8)

Sum of denominators of L.H.S. 

= x – 4 + x – 6 

= 2x – 10 …(i) 

Sum of denominators of R.H.S. 

= x – 2 + x – 8 

= 2x – 10 …(ii) 

(i) and (ii) are same so by formula 

2x – 10 = 0 

⇒ 2x = 10 

⇒ x = 5