Begin by cross-multiplying.

Distribute the on the left side and expand the polynomial on the right.

Combine like terms and rearrange to set the equation equal to zero.

Now we can isolate and solve for by adding to both sides.

With rational equations we must first note the domain, which is all real numbers except  and . That is, these are the values of  that will cause the equation to be undefined. Since the least common denominator of , , and  is , we can mulitply each term by the LCD to cancel out the denominators and reduce the equation to . Combining like terms, we end up with . Dividing both sides of the equation by the constant, we obtain an answer of . However, this solution is NOT in the domain. Thus, there is NO SOLUTION because  is an extraneous answer. 

Factor out  from the numerator which gives us

Hence we get the following

which is equal to

First we convert each of the denominators into an LCD which gives us the following:

Now we add or subtract the numerators which gives us:

Simplifying the above equation gives us the answer which is:

The two fractions on the left side of the equation need a common denominator. We can easily do find one by multiplying both the top and bottom of each fraction by the denominator of the other.

   becomes .

  becomes .

Now add the two fractions:

To solve, multiply both sides of the equation by , yielding

 .

Multiply both sides by 3:

Move all terms to the same side:

This looks like a complicated equation to factor, but luckily, the only factors of 37 are 37 and 1, so we are left with

 .

Our solutions are therefore

and

.

Multiply both sides by :

Factor this using the -method. We split the middle term using two integers whose sum is  and whose product is . These integers are :

Set each factor equal to 0 and solve separately:

 or

The first step of problems like this is to try to factor the quadratic and see if it shares a factor with the linear polynomial in the denominator. And as it turns out,

So our rational function is equal to 

which is as simplified as it can get.

When dividing fractions, we multiply by the reciprocal of the second fraction. Therefore, the problem becomes:

Our final unfactored expression is therefore .

Since the two fractions already have a common denominator, you can add the fractions by adding up the two numerators and keeping the common denominator:

Next you will algebraically solve for  by isolating it on one side of the equation. The first step is to multiply each side by :

Cancel out the  on the left and distribute out on the right. Then solve for  by subtracting  to the left and subtracting 10 to the right. Finally divide each side by negative 2:

With rational equations we must first note the domain, which is all real numbers except . (If , then the term  will be undefined.) Next, the least common denominator is , so we multiply every term by the LCD in order to cancel out the denominators. The resulting equation is . Subtract  on both sides of the equation to collect all variables on one side: . Lastly, divide by the constant to isolate the variable, and the answer is . Be sure to double check that the solution is in the domain of our equation, which it is.