Help Solving These?

There Are more like these ,
but i need examples to help me with the others.

1.5/2y+6 + 1/y-2=3/y+3

2. 3/3x-2 = 4/2x+1

3.5/3x - 1/9 = 1/x

4.2/x^2-x-12 -6/x+4 = 2/x-3

You will need to use the distributive property of multiplication.

5/(2y+6) + 1/(y-2) = 3/(y+3)
5/2(y+3) + 1/(y-2) = 3/(y+3)
2(5)/(y+3) + 2/(y-2) = 2(3)/(y+3)
10/(y+3) + 2/(y-2) = 6/(y+3)
2/(y-2) = -4/(y+3)
(y-2)(2)/(y-2) = -4(y-2)/(y+3)
2 = -4(y-2)/(y+3)
2/(-4) = -4(y-2)/-4(y+3)
-(1/2) = (y-2)/(y+3)
-(1/2)(y+3) = (y+3)(y+2)/(y+3)
-(1/2)(y+3) = (y+2)
-(1/2)y - 3/2 = y + 2
-(3/2) - 2 = y + (1/2)y
-(3/2) - 4/2 = (2/2)y + (1/2)y
-(7/2) = (3/2)y
-(7/2) / (3/2) = y
- (7/3) = y
y = - 2 1/3

hmmm. I don't get that.

5/(2y+6) + 1/(y-2) = 3/(y+3)
putting all over 2(y+3)(y-2), and discarding the denominator, we have

5(y-2) + 2(y+3) = 3*2(y-2)
5y - 10 + 2y + 6 = 6y - 12
y = -8

check:

5/-10 + 1/-10 = 6/-10 = 3/-5
----------------------------
3/(3x-2) = 4/(2x+1)
putting all over (3x-2)(2x+1), we have
3(2x+1) = 4(3x-2)
6x + 3 = 12x - 8
6x = 11
x = 11/6
-------------------------------
5/3x - 1/9 = 1/x
put all over 9x:
5*3 - x = 9
x = 6
------------------------------
2/(x^2-x-12) -6/(x+4) = 2/(x-3)
note that x^2-x-12 = (x-4)(x+3)
So, putting all over (x-4)(x+4)(x-3)(x+3), we have
2(x+4)(x-3) - 6(x+4)(x-3)(x+3) = 2(x-4)(x+4)(x+3)

Now, that's kind of nasty. I think you have a typo. If instead we have

2/(x^2+x-12) -6/(x+4) = 2/(x-3)
the LCD = (x+4)(x-3), and we get

2 - 6(x-3) = 2(x+4)
2 - 6x + 18 = 2x + 8
x = 3/2

Sure! Let's go through each of these equations step by step to solve them.

1. 5/(2y+6) + 1/(y-2) = 3/(y+3)

First, let's simplify the equation by finding a common denominator for the fractions. The common denominator in this case is (2y+6)(y-2)(y+3).

Multiplying each fraction by the appropriate factor to get the common denominator, we have:
(5(y-2) + (2y+6))/((2y+6)(y-2)) + (1*(y+3))/((y-2)(y+3)) = (3*(2y+6))/((y+3)(y-2))

Simplifying further, we get:
(5y - 10 + 2y + 6)/((2y+6)(y-2)) + (y + 3)/((y-2)(y+3)) = (6y + 18)/((y+3)(y-2))

Combining like terms, we have:
(7y - 4)/((2y+6)(y-2)) + (y + 3)/((y-2)(y+3)) = (6y + 18)/((y+3)(y-2))

Now, let's multiply both sides of the equation by (2y+6)(y-2)(y+3) to eliminate the denominators:

[(7y - 4)(y-2)(y+3)] + [(y + 3)(2y+6)(y-2)] = [(6y + 18)(2y+6)(y-2)]

Expanding and simplifying, we get:
(7y - 4)(y^2 + y - 6) + (y + 3)(2y+6)(y-2) = (6y + 18)(2y+6)(y-2)

Now, we have a quadratic equation. Let's multiply it out and move all terms to one side to solve for y. After simplifying, we get:

y^3 - 11y^2 + 45y + 8 = 0

Now, you can solve this cubic equation using numerical methods or factoring techniques.

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

To solve this equation, first, let's cross-multiply and get rid of the denominators:

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

Expanding and simplifying, we have:
6x + 3 = 12x - 8

Next, let's isolate the variable by moving all terms with x to one side:

6x - 12x = -8 - 3

Simplifying further, we get:
-6x = -11

Finally, divide both sides by -6 to solve for x:

x = -11/-6

Therefore, the solution is x = 11/6.

3. 5/(3x) - 1/9 = 1/x

To solve this equation, let's first find a common denominator for the fractions. The common denominator in this case is 9x.

Multiplying each fraction by the appropriate factor to get the common denominator, we have:

(5*(3))/(3x) - (1*(x))/(9) = (1*(3))/(x)

Simplifying, we get:
15/3x - x/9 = 3/x

Next, let's cross-multiply and eliminate the denominators:

(15)(9) - (x)(3x) = (3)(3x)

Expanding and simplifying, we have:
135 - 3x^2 = 9x

Moving all terms to one side, we get a quadratic equation:
3x^2 + 9x - 135 = 0

Now, you can solve this quadratic equation using factoring, the quadratic formula, or numerical methods.

4. 2/(x^2-x-12) - 6/(x+4) = 2/(x-3)

To solve this equation, let's first find a common denominator for the fractions. The common denominator in this case is (x-3)(x+4).

Multiplying each fraction by the appropriate factor to get the common denominator, we have:

(2*(x-3)) - 6(x-3)(x+4) = 2(x+4)

Simplifying, we get:
2x - 6 - 6x^2 + 30x - 72 = 2x + 8

Next, let's combine like terms and move all terms to one side:

-6x^2 + 30x + 2x - 30x - 2x - 6 - 72 - 8 = 0

Simplifying further, we have:
-6x^2 - 78 = 0

Now, you can solve this quadratic equation using factoring, the quadratic formula, or numerical methods.

I hope this helps you solve these equations! If you have any more questions, feel free to ask.