Verified answer

The first problem has been cut off by Yahoo format because there need to be spaces between the problem itself. If something is too long, Yahoo cuts off the end of that line.

Solve

(3x)/(x-2)=4+(x)/(5)

what's the restriction?

Find the LCD:

(3x)/((x-2)) + 4 + (x)/(5)

5(x - 2) is the LCD here.

15x = x^2 + 18x - 40

x^2 + 18x - 40 = 15x

x^2 + 3x - 40 = 0

x^2 + 3x - 40

(x + 8)(x - 5) = 0

x + 8 = 0

x - 5 = 0

x = -8

x = 5

Restrictions are x ≠ 2, x ≠ -5

Those would be division by zero.

Subtract

(3x-6)/(x^(2)+x-6) = (-x+2)/(x^(2)+x-6)

Solution and restriction?

Factor out a GCF and go from there:

(3(x-2))/((x+3)(x-2)) = (-x+2)/(x^2+x-6)

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

3(x - 2) = -x+2

(3x - 6) = -x+2

4x - 6 = 2

4x = 8

x = 2

This is actually "no solution" because in the original problem, x can't equal 2 or -3 because of division by zero.

Divide

(3y-12)/(2y+4) / (6y^(3)-24y^(2))/(4y+8) state any restrictions on the variables

solution and restriction?

Factor out the GCF of 3:

((((3(y-4)))/(2y+4))) / (6y^3 - 24y^2)/(4y+8)

(((3(y-4))/(2(y+2)))) / (6y^3 - 24y^2)/(4y+8)

(((1)/(6y^2(y-4))*(3(y-4)) / (2(y+2))))/(4y+8)

((1)/(2y^2) * (1)/(2(y+2))) / (4y+8)

((1)/(4y^2(y+2))) / (4y+8)

(1)/(4(y+2)) * (1)/(4y^2(y+2))

(1)/(16y^2(y+2)(y+2))

1 / (16y^2(y+2)^2)

Look at the values in the factored out portions that would have division by zero, and that will be the restrictions.