A **rational equation** is an equation with rational expressions on either side of the equals sign.

**ONE TECHNIQUE** for solving rational equations is cross-multiplication — what some textbooks call the means/extremes property.

This method works only if on each side of the equation there is only one rational expression.

**Example 1:**

Solve:

Cross multiplying, we get:

*x*^{2} + 2*x* = 7*x* + 14

This quadratic equation can be solved by factoring.

* x*^{2} – 5*x* – 14 = 0

(*x* – 7)(*x* + 2) = 0

**Remember** to check in the original equation for validity of solutions. In this case, *x* = 7 is valid but *x* = **–**2 isn't, since it means division by zero in the original equation.

**ANOTHER METHOD** is to multiply through by the least common denominator of all of the fractions on either side of the equation.

**Example 2:**

Solve:

The least common denominator (LCD) in this case is 16*x*. So, multiply both sides of the equation by 16*x*.

* x*^{2} – 6 = 5*x*

Solve the quadratic equation by factoring.

* x*^{2} – 5*x *– 6 = 0

* ** *(*x* – 6)(*x* + 1) = 0

* x* = 6 or *x* = –1.

**Remember** to check back to make sure these solutions are valid – that is, that they don't result in division by zero when substituted in the original equation. In this case, both solutions are valid.