To solve the equations, we will first simplify the expressions on both sides of the equation and then solve for x.

a) 30/x - 30/(x+2) = 1/2

To simplify the left side of the equation, we need to find a common denominator. The common denominator is x(x+2).

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

Now, we can combine the fractions:

(30x + 60 - 30x)/(x(x+2)) = 1/2

Simplifying further:

60/(x(x+2)) = 1/2

To get rid of the fraction, we can multiply both sides of the equation by 2:

2 * 60/(x(x+2)) = 2 * 1/2

120/(x(x+2)) = 1

Now, we can cross-multiply:

120 = x(x+2)

Expanding the equation:

120 = x^2 + 2x

Rearranging the equation:

x^2 + 2x - 120 = 0

Now, we can solve this quadratic equation by factoring or using the quadratic formula. Factoring, we have:

(x + 12)(x - 10) = 0

Setting each factor equal to zero:

x + 12 = 0 or x - 10 = 0

Solving for x:

x = -12 or x = 10

Therefore, the solutions to the equation are x = -12 and x = 10.

b) 30/(x+2) - 30/x = 1/2

To simplify the left side of the equation, we need to find a common denominator. The common denominator is x(x+2).

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

Now, we can combine the fractions:

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

Simplifying further:

(30x - 30x - 60)/(x(x+2)) = 1/2

-60/(x(x+2)) = 1/2

To get rid of the fraction, we can multiply both sides of the equation by 2:

2 * -60/(x(x+2)) = 2 * 1/2

-120/(x(x+2)) = 1

Now, we can cross-multiply:

-120 = x(x+2)

Expanding the equation:

-120 = x^2 + 2x

Rearranging the equation:

x^2 + 2x + 120 = 0

Now, we can solve this quadratic equation by factoring or using the quadratic formula. Factoring, we have:

(x + 10)(x + 12) = 0

Setting each factor equal to zero:

x + 10 = 0 or x + 12 = 0

Solving for x:

x = -10 or x = -12

Therefore, the solutions to the equation are x = -10 and x = -12.