**Question 13. (7x – 2)/(5x – 1) = (7x + 3)/(5x + 4)**

**Solution:**

Given:

(7x – 2) / (5x – 1) = (7x +3)/(5x + 4)

(7x – 2) / (5x – 1) – (7x +3)/(5x + 4) = 0

By taking LCM as (5x – 1) (5x + 4)

((7x-2) (5x+4) – (7x+3)(5x-1)) / (5x – 1) (5x + 4) = 0

After cross-multiplying we will get,

(7x-2) (5x+4) – (7x+3)(5x-1) = 0

Now after simplification,

35x

^{2}+ 28x – 10x – 8 – 35x^{2}+ 7x – 15x + 3 = 010x – 5 = 0

10x = 5

x = 5/10

x = 1/2

Now verify the given equation by substituting x = 1/2,

(7x – 2) / (5x – 1) = (7x +3)/(5x + 4)

x = 1/2

(7(1/2) – 2) / (5(1/2) – 1) = (7(1/2) + 3) /(5(1/2) + 4)

(7/2 – 2) / (5/2 – 1) = (7/2 + 3) / (5/2 + 4)

((7-4)/2) / ((5-2)/2) = ((7+6)/2) / ((5+8)/2)

(3/2) / (3/2) = (13/2) / (13/2)

1 = 1

Here, L.H.S. = R.H.S.,

Thus the given equation is verified.

**Question 14. ((x+1)/(x+2))**^{2} = (x+2)/(x + 4)

^{2}= (x+2)/(x + 4)

**Solution:**

Given:

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

^{2}= (x+2) / (x + 4)(x+1)

^{2}/ (x+2)^{2}– (x+2) / (x + 4) = 0By taking LCM as (x+2)

^{2}(x+4)((x+1)

^{2}(x+4) – (x+2) (x+2)^{2}) / (x+2)^{2}(x+4) = 0After cross-multiplying we will get,

(x+1)

^{2}(x+4) – (x+2) (x+2)^{2}= 0Now expand the equation as follows,

(x

^{2}+ 2x + 1) (x + 4) – (x + 2) (x^{2}+ 4x + 4) = 0x

^{3}+ 2x^{2}+ x + 4x^{2}+ 8x + 4 – (x^{3}+ 4x^{2}+ 4x + 2x^{2}+ 8x + 8) = 0x

^{3}+ 2x^{2}+ x + 4x^{2}+ 8x + 4 – x^{3}– 4x^{2}– 4x – 2x^{2}– 8x – 8 = 0-3x – 4 = 0

x = -4/3

Now verify the given equation by substituting x = -4/3,

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

^{2}= (x+2) / (x + 4)x = -4/3

(x+1)

^{2}/ (x+2)^{2}= (x+2) / (x + 4)(-4/3 + 1)

^{2}/ (-4/3 + 2)^{2}= (-4/3 + 2) / (-4/3 + 4)((-4+3)/3)

^{2}/ ((-4+6)/3)^{2}= ((-4+6)/3) / ((-4+12)/3)(-1/3)

^{2}/ (2/3)^{2}= (2/3) / (8/3)1/9 / 4/9 = 2/3 / 8/3

1/4 = 2/8

1/4 = 1/4

Here, L.H.S. = R.H.S.,

Thus, the given equation is verified.

**Question 15. ((x+1)/(x-4))**^{2} = (x+8)/(x-2)

^{2}= (x+8)/(x-2)

**Solution:**

Given:

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

^{2}= (x+8)/(x-2)(x+1)

^{2}/ (x-4)^{2}– (x+8) / (x-2) = 0By taking LCM as (x-4)

^{2}(x-2)((x+1)

^{2}(x-2) – (x+8) (x-4)^{2}) / (x-4)^{2}(x-2) = 0After cross-multiplying we will get,

(x+1)

^{2}(x-2) – (x+8) (x-4)^{2}= 0After expansion we get,

(x

^{2}+ 2x + 1) (x-2) – ((x+8) (x^{2}– 8x + 16)) = 0x

^{3}+ 2x^{2}+ x – 2x^{2}– 4x – 2 – (x^{3}– 8x^{2}+ 16x + 8x^{2}– 64x + 128) = 0x

^{3}+ 2x^{2}+ x – 2x^{2}– 4x – 2 – x^{3}+ 8x^{2}– 16x – 8x^{2}+ 64x – 128 = 045x – 130 = 0

x = 130/45

x = 26/9

Now verify the given equation by substituting x = 26/9

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

^{2}= (x+8)/(x-2)(x+1)

^{2}/ (x-4)^{2}= (x+8) / (x-2)x = 26/9

(26/9 + 1)

^{2}/ (26/9 – 4)^{2}= (26/9 + 8) / (26/9 – 2)((26+9)/9)

^{2}/ ((26-36)/9)^{2}= ((26+72)/9) / ((26-18)/9)(35/9)

^{2}/ (-10/9)^{2}= (98/9) / (8/9)(35/-10)

^{2}= (98/8)(7/2)

^{2}= 49/449/4 = 49/4

Here, L.H.S. = R.H.S.,

Thus, the given equation is verified.

**Question 16. (9x-7)/(3x+5) = (3x-4)/(x+6)**

**Solution:**

Given:

(9x-7)/(3x+5) = (3x-4)/(x+6)

(9x-7)/(3x+5) – (3x-4)/(x+6) = 0

By taking LCM as (3x+5) (x+6)

((9x-7) (x+6) – (3x-4) (3x+5)) / (3x+5) (x+6) = 0

After cross-multiplying we will get,

(9x-7) (x+6) – (3x-4) (3x+5) = 0

Upon expansion we will get,

9x

^{2}+ 54x – 7x – 42 – (9x^{2}+ 15x – 12x – 20) = 044x – 22 = 0

44x = 22

x = 22/44

= 2/4

x = 1/2

Now verify the given equation by substituting x =1/2,

(9x-7)/(3x+5) = (3x-4)/(x+6)

x = 1/2

(9(1/2) – 7) / (3(1/2) + 5) = (3(1/2) – 4) / ((1/2) + 6)

(9/2 – 7) / (3/2 + 5) = (3/2 – 4) / (1/2 + 6)

((9-14)/2) / ((3+10)/2) = ((3-8)/2) / ((1+12)/2)

-5/2 / 13/2 = -5/2 / 13/2

-5/13 = -5/13

Here, L.H.S. = R.H.S.,

Thus, the given equation is verified.

**Question 17. (x+2)/(x+5) = x/(x+6)**

**Solution:**

Given:

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

(x+2)/(x+5) – x/(x+6) = 0

By taking LCM as (x+5) (x+6)

((x+2) (x+6) – x(x+5)) / (x+5) (x+6) = 0

After cross-multiplying we will get,

(x+2) (x+6) – x(x+5) = 0

Upon expansion we will get

x

^{2}+ 8x + 12 – x^{2}– 5x = 03x + 12 = 0

3x = -12

x = -12/3

x = -4

Now verify the given equation by substituting x = -4,

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

x = -4

(-4 + 2) / (-4 + 5) = -4 / (-4 + 6)

-2/1 = -4 / (2)

-2 = -2

Here, L.H.S. = R.H.S.,

Thus, the given equation is verified.

**Question 18. 2x – (7-5x)/9x – (3+4x) = 7/6**

**Solution:**

Given:

2x – (7-5x) / 9x – (3+4x) = 7/6

(2x – 7 + 5x) / (9x – 3 – 4x) = 7/6

(7x – 7) / (5x – 3) = 7/6

After cross-multiplying we will get,

6(7x – 7) = 7(5x – 3)

42x – 42 = 35x – 21

42x – 35x = -21 + 42

7x = 21

x = 21/7

x = 3

Now verify the given equation by substituting

2x – (7-5x) / 9x – (3+4x) = 7/6

(7x – 7) / (5x – 3) = 7/6

x = 3

(7(3) -7) / (5(3) – 3) = 7/6

(21-7) / (15-3) = 7/6

14/12 = 7/6

7/6 = 7/6

Here, L.H.S. = R.H.S.,

Thus, the given equation is verified.

**Question 19. (15(2-x) – 5(x+6))/(1-3x) = 10**

**Solution:**

Given:

15(2-x) – 5(x+6) / (1-3x) = 10

(30-15x) – (5x + 30) / (1-3x) = 10

After cross-multiplying we will get,

(30-15x) – (5x + 30) = 10(1- 3x)

30- 15x – 5x – 30 = 10 – 30x

30- 15x – 5x – 30 + 30x = 10

10x = 10

x = 10/10

x = 1

Now verify the given equation by substituting x =1,

(15(2-x) – 5(x+6)) / (1-3x) = 10

x = 1

(15(2-1) – 5(1+6)) / (1- 3) = 10

(15 – 5(7))/-2 = 10

(15-35)/-2 = 10

-20/-2 = 10

10 = 10

Here, L.H.S. = R.H.S.,

Thus, the given equation is verified.

**Question 20. (x+3)/(x-3) + (x+2)/(x-2) = 2**

**Solution:**

Given:

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

By taking LCM as (x-3) (x-2)

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

After cross-multiplying we will get,

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

Upon expansion we will get,

x

^{2}+ 3x – 2x – 6 + x^{2}– 3x + 2x – 6 = 2(x^{2}– 3x – 2x + 6)2x

^{2}– 12 = 2x^{2}– 10x + 122x

^{2}– 2x^{2}+ 10x = 12 + 1210x = 24

x = 24/10

x = 12/5

Now verify the given equation by substituting x = 12/5,

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

x = 12/5

(12/5 + 3)/(12/5 – 3) + (12/5 + 2)/(12/5 – 2) = 2

((12+15)/5)/((12-15)/5) + ((12+10)/5)/((12-10)/5) = 2

(27/5)/(-3/5) + (22/5)/(2/5) = 2

-27/3 + 22/2 = 2

((-27×2) + (22×3))/6 = 2

(-54 + 66)/6 = 2

12/6 = 2

2 = 2

Here, L.H.S. = R.H.S.,

Thus, the given equation is verified.

**Question 21. ((x+2)(2x-3) – 2x**^{2} + 6)/(x-5) = 2

^{2}+ 6)/(x-5) = 2

**Solution:**

We have,

((x+2) (2x-3) – 2x

^{2}+ 6)/(x-5) = 2After cross-multiplying we will get,

(x+2) (2x-3) – 2x

^{2}+ 6) = 2(x-5)2x

^{2}– 3x + 4x – 6 – 2x^{2}+ 6 = 2x – 10x = 2x – 10

x – 2x = -10

-x = -10

x = 10

Now verify the given equation by substituting x = 10

((x+2) (2x-3) – 2x

^{2}+ 6)/(x-5) = 2x = 10

((10+2) (2(10) – 3) – 2(10)

^{2}+ 6)/ (10-5) = 2(12(17) – 200 + 6)/5 = 2

(204 – 194)/5 = 2

10/5 = 2

2 = 2

Here, L.H.S. = R.H.S.,

Thus, the given equation is verified.

**Question 22. (x**^{2} – (x+1)(x+2))/(5x+1) = 6

^{2}– (x+1)(x+2))/(5x+1) = 6

**Solution:**

Given:

(x

^{2}– (x+1) (x+2))/(5x+1) = 6After cross-multiplying we will get,

(x

^{2}– (x+1) (x+2)) = 6(5x+1)x

^{2}– x^{2 }– 2x – x – 2 = 30x + 6-3x – 2 = 30x + 6

30x + 3x = -2 – 6

33x = -8

x = -8/33

Now verify the given equation by substituting x = -8/33

(x

^{2}– (x+1) (x+2))/(5x+1) = 6x = -8/33

((-8/33)

^{2}– ((-8/33)+1) (-8/33 + 2))/(5(-8/33)+1) = 6(64/1089 – ((-8+33)/33) ((-8+66)/33)) / (-40+33)/33) = 6

(64/1089 – (25/33) (58/33)) / (-7/33) = 6

(64/1089 – 1450/1089) / (-7/33) = 6

((64-1450)/1089 / (-7/33)) = 6

-1386/1089 × 33/-7 = 6

1386 × 33 / 1089 × -7 = 6

6 = 6

Here, L.H.S. = R.H.S.,

Thus, the given equation is verified.

**Question 23. ((2x+3) – (5x-7))/(6x+11) = -8/3**

**Solution:**

Given:

((2x+3) – (5x-7))/(6x+11) = -8/3

After cross-multiplying we will get,

3((2x+3) – (5x-7)) = -8(6x+11)

3(2x + 3 – 5x + 7) = -48x – 88

3(-3x + 10) = -48x – 88

-9x + 30 = -48x – 88

-9x + 48x = -88 – 30

39x = -118

x = -118/39

Now verify the given equation by substituting x = -118/39

((2x+3) – (5x-7))/(6x+11) = -8/3

x = -118/39

((2(-118/39) + 3) – (5(-118/39) – 7)) / (6(-118/39) + 11) = -8/3

((-336/39 + 3) – (-590/39 – 7)) / (-708/39 + 11) = -8/3

(((-336+117)/39) – ((-590-273)/39)) / ((-708+429)/39) = -8/3

(-219+863)/39 / (-279)/39 = -8/3

644/-279 = -8/3

-8/3 = -8/3

Here, L.H.S. = R.H.S.,

Thus, the given equation is verified.

**Question 24. Find the positive value of x for which the given equation is satisfied:**

**(i) (x**^{2} – 9)/(5+x^{2}) = -5/9

^{2}– 9)/(5+x

^{2}) = -5/9

**Solution:**

Given:

(x2 – 9)/(5+x2) = -5/9

After cross-multiplying we will get,

9(x

^{2}– 9) = -5(5+x^{2})9x

^{2 }– 81 = -25 – 5x^{2}9x

^{2}+ 5x^{2}= -25 + 8114x

^{2}= 56x

^{2}= 56/14x

^{2}= 4x = √4

x = 2

**(ii) (y**^{2} + 4)/(3y^{2} + 7) = 1/2

^{2}+ 4)/(3y

^{2}+ 7) = 1/2

**Solution:**

Given:

(y

^{2}+ 4)/(3y^{2}+ 7) = 1/2After cross-multiplying we will get,

2(y

^{2}+ 4) = 1(3y^{2}+ 7)2y

^{2}+ 8 = 3y^{2}+ 73y

^{2}– 2y^{2}= 7 – 8y

^{2}= -1y = √-1

= 1