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3)

4/(x+1) = 3/(x+2)

=> 4(x+2) = 3(x+1)

=> 4x + 8 = 3x + 3

=> x + 8 = 3

=> x = -5

4)

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

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

=> x^2 = x^2 + x - 6

=> 0 = x - 6

=> x = 6

28)

1/3 + 2/x = 3/x

Use fraction busters, find the LCD (LCM). lcd = 3x. Multiply each term by 3x, things cancel.

=> 3x * [1/3 + 2/x = 3/x]

=> x + 6 = 9

=> x = 3

30)

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

LCD = 2x(x+1)

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

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

=> 6x - 2x - 2 = x^2 + x

=> 4x - 2 = x^2 + x

=> 0 = x^2 - 3x + 2

=> 0 = (x - 1)(x - 2)

Use zero-product rule

=> x - 1 = 0 or x - 2 = 0

=> x = 1 or x = 2

3) Multiply both sides by the product of the denominators, (x + 1)(x + 2):

4(x + 2) = 3(x + 1)

4x + 8 = 3x + 3

x = -5

4) Same advice as in #3, multiply by x(x + 3):

x² = x² + x - 6

x = 6

28) Multiply by the product of all the (different) denominators: 3x

x + 6 = 9

x = 3

30) Same advice as #28, multiply by 2x(x + 1):

6x - 2(x + 1) = x² + x

x² - 3x + 2 = 0

(x - 1)(x - 2) = 0

x = {1, 2}

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

Cross multiply.

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

x^2 = x^2 +3x -2x -6 = x^2 +x -6

The x^2 terms cancel.

x-6 = 0, x=6

30. 3/(x+1) - 1/x = 1/2

Multiply both sides by 2x(x+1) to clear the denominators

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

6x - 2x - 2 = x^2+x

Combining terms,

x^2 -3x + 2 =0

(x-2)(x-1)=0

x = 2 or x = 1

3. 4 / (x + 1) = 3/(x + 2)

4 = 3(x + 1) / (x + 2) multiply each side by x + 1

4(x + 2) = 3(x + 1) multiply each side by x + 2

4x + 8 = 3x + 3 apply the distributive property of multiplication over addition to both sides

4x + 8 - 3x = 3 subtract 3x from each side

4x - 3x = 3 - 8 subtract 8 from each side

x = -5 simplify each side