Section 1.1 - Rational Equations that Reduce to Linear

Section Objectives

Recognize an expression as rational.
Determine the values of the variable that are restricted from a rational expression.
Solve special cases of rational equations that reduce to linear equations.

A rational expression is an algebraic expression in which whole number powers of the variable may appear in the numerator or denominator of a fraction.

For example, these are rational expressions:

$\displaystyle \frac{3x}{x+5},$

$\displaystyle \frac{x^2+x}{3x^2+1},$

$8x-7$

For now, we will be most interested in rational expressions that involve only first powers of the variable.

Restricted Values of the Variable

Recall that division by zero is not defined. Since rational expressions may contain variables in the denominator, we must be careful to avoid any values for the variable that would result in a zero denominator. We'll call these values restricted values.

For example...

$x=2$ $\displaystyle \frac{5x}{2x-4}$ . Do you see why?

$r$ $\displaystyle \frac{r-3}{3r-15}$ ?

Solving Rational Equations that Reduce to Linear

We will study lots of rational expressions and rational equations throughout the semester. Right now, we are only interested in rational equations that can be reduced to linear equations. We'll look at two special cases:

rational equations that are proportions and can be solved by cross multiplying, and
rational equations in which fractions can be cleared by multiplication.

Important point: A solution of a rational equation can never be a restricted value. Be sure to check for this!

Type 1 Examples

$x$ $\qquad \displaystyle \frac{-6}{x} = \frac{-15}{x-3}$ .

$v$ $\qquad \displaystyle \frac{1}{v-5}=\frac{2}{3v-15}$

Type 2 Examples

$w$ $\qquad \displaystyle \frac{w}{w-4} = \frac{4}{w-4}-\frac{4}{5}$

$u$ $\qquad \displaystyle \frac{u-1}{4u}+\frac{1}{6} = \frac{1}{u}$

$x$ $\qquad \displaystyle -\frac{5}{4x-16}+1=-\frac{2}{x-4}$

What about solving a formula?

$s$ $\qquad \displaystyle \frac{L}{p(r+s)}=n$