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:

$\frac{7}{x\text{\hspace{0.17em}}+\text{\hspace{0.17em}}2}=\frac{x}{x\text{\hspace{0.17em}}+\text{\hspace{0.17em}}2}$

Cross multiplying, we get:

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

This quadratic equation can be solved by factoring.

${x}^{2}-5x-14=0$

$(x-7)(x+2)=0$

**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:

$\frac{x}{16}-\frac{3}{8x}=\frac{5}{16}$

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

$\frac{x\left(16x\right)}{16}-\frac{3\left(16x\right)}{8x}=\frac{5\left(16x\right)}{16}$

${x}^{2}-6=5x$

Solve the quadratic equation by factoring.

${x}^{2}-5x-6=0$

$(x-6)(x+1)=0$

$x=6\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{or}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x=-1$

**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.