Rational equations are equations made up of rational expressions.  Just as rational numbers are numbers that can be expressed as fractions, rational expressions can also be expressed as fractions.

Here are some examples of rational equations:

Since these fractions have the same denominator, it must be that x = 4.

There are some options in how to find the value of x here.  We can find a common denominator; in this case the common denominator is 6.  We convert 2/3 to 4/6, and have 4/6 = (x+1)/6.  Now we can see that 4 = x+1, so x = 3:

Another way is to take the cross product, as we do when solving proportions; as we call, the cross product property states that the product of the means = the product of the extremes.   So we multiply 2*6 and set it equal to 3(x + 1)  (remember that x + 1 is a quantity, so include the parentheses when cross multiplying):

2*6 = 3(x + 1)

12 = 3x + 3

9 = 3x

x = 3

A third way to approach solving this equation is to multiply both sides by the common denominator of 6 which will eliminate the denominators:

This is a bit more complex than the other examples and is an advanced topic, but if we remember how to add fractions when they are numeric we can approach this problem.  When we add numeric fractions, we must have a common denominator.   One way to find a common denominator is to multiply the denominators.  We will do the same thing here.

The denominators are x, x+2, and 3x.  If we multiply the first two denominators, we get a common denominator of x(x+2).  The common denominator between x(x+2) and 3x is 3x(x+2), which is our common denominator for the three expressions.

Now we convert our fractions to all have this common denominator of 3x(x+2). 

To convert 1/x, we determine what we multiply x by to get 3x(x+2) (that is, we use polynomial division to divide 3x(x+2) by x).  This quotient is 3(x+2), so we multiply the denominator by 3(x+2) and we multiply the numerator by the same thing (note that you can verify by reducing the equivalent fraction - you should get back what you started with): 

To convert 2/(x+2), we would multiply the denominator by 3x so we multiply the numerator by 3x:

To convert 1/3x, we would multiply the denominator by (x+2) so we multiply the numerator by (x+2):

Now that we converted all our rational expressions to have a common denominator, the equation changes:

At this point we can see that we just need to work with the numerators, as the denominators don't really matter any more.  To make this point clearer, let's multiply both sides of the equation by the common denominator of 3x(x+2), which will eliminate the denominator (just as we did in the second example above):

Now we can solve this equation:

3(x+2) + 6x = x + 2

3x + 6 + 6x = x + 2     (clear parentheses)

9x + 6 = x + 2             (combine like terms)

8x = -4                         (subtract x from both sides; subtract 6 from both sides)

x = -1/2                       (divide both sides by 8)

As always, solutions may be verified by plugging the value back into the original equation; if we have equality, the solution is correct.