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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...Show more

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}...Show more

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...Show more

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...Show more