# Solving Equations with Rational Expressions; Problems Involving Proportions

## Rational expressions, like proportions, are extremely useful applications of algebra, that can be solved using simple algebraic techniques.

#### Key Points

• A rational equation means that you are setting two rational expressions equal to each other. Proportions are perfect examples of a rational expression. Even if they look different, they can be simplified down into the same expression:$(\frac 2 4)= (\frac 1 2)$.

• If you have a rational equation where the denominators are the same, then the numerators must be the same. This in turn suggests a strategy: find a common denominator, and then set the numerators equal using algebraic techniques.

• Remember, all normal algebraic rules apply to solving rational equations. Such as, you still can not divide by 0.

#### Terms

• An expression that can be expressed as the quotient of two polynomials.

#### Examples

• When given the rational equation: $(\frac a b)=(\frac c d)$This can be solved by either finding a common denominator, or by setting it up like:$ad=cb$and then solving it algebraically.

#### Figures

1. ##### Interactive Graph: Graphical Determination of x(x-30)=0

To determine the solutions to the equation $x(x-30)=0$, graph it and look for where the dependent variable crosses the x-axis. The graph crosses the x-axis at 0 and 30, the solution to the equation.

A rational equation means that you are setting two rational expressions equal to each other. The goal is to solve for x; that is, find the value(s) that make the equation true.

Suppose you are told that:

$\frac x 5 = \frac 3 5$

If you think about it, the x in this equation has to be a 3. That is to say, if $x=3$ then this equation is true; for any other x value, this equation is false.If this is not so apparent to you, you can always solve it the old fashioned way, by working it out. Start by isolating the variable you are solving for:$x = (\frac 3 5)5$ which simplifies down to $x = 3$

This leads us to a very general rule: If you have a rational equation where the denominators are the same, then the numerators must be the same.

This in turn suggests a strategy: find a common denominator, and then set the numerators equal.

For example, consider the rational equation $\frac 3 {x^2+12x+36} = \frac {4x}{x^3+4x^2-12x}$

by factoring the denominators,we find that we must multiply the left side of the equation by $\frac {x(x-2)}{x(x-2)}$ and the right side of the equation by $\frac {x+6}{x+6}$, giving

$\frac {3x(x-2)}{(x+6)^2x(x-2)}=\frac {4x(x+6)}{x(x+6)^2(x-2)}$

Based on the rule above—since the denominators are equal, we can now assume the numerators are equal, so we know that $3x(x-2)= 4x(x+6)$or, multiplied out, that $3x^2-6x=4x^2+24x$

What we’re dealing with, in this case, is a quadratic equation. As always, move everything to one side, giving $x^2+30x=0$

and then factor. A common mistake in this kind of problem is to divide both sides by x; this loses one of the two solutions.

$x(x-30)=0$

Two solutions to the quadratic equation. However, in this case, $x=0$ is not valid, since it was not in the domain of the original right-hand fraction. (Why?) So this problem actually has only one solution, $x=–30$.This is shown in Figure 1

As always, it is vital to remember what we have found here. We started with the equation $\frac {3x(x-2)}{(x+6)^2x(x-2)}=\frac {4x(x+6)}{x(x+6)^2(x-2)}$. We have concluded now that if you plug $x=–30$ into that equation, you will get a true equation (you can verify this on your calculator). For any other value, this equation will evaluate false.

#### Key Term Glossary

algebraic
or function}} Containing only numbers, letters and arithmetic operators.
##### Appears in these related concepts:
denominator
The number or expression written below the line in a fraction (thus 2 in ½).
##### Appears in these related concepts:
domain
A small magnetized area of a bubble storing one bit of memory, a bubble.
##### Appears in these related concepts:
equation
An assertion that two expressions are equal, expressed by writing the two expressions separated by an equals sign. E.g. x=5.
##### Appears in these related concepts:
expression
An arrangement of symbols denoting values, operations performed on them, and grouping symbols. E.g. (2x+4)/2
##### Appears in these related concepts:
factor
To find all the factors of (a number or other mathematical object) (the objects that divide it evenly).
##### Appears in these related concepts:
fraction
A ratio of two numbers, the numerator and the denominator, usually written one above the other and separated by a horizontal bar.<!--rational number (a rational number can always be expressed as a fraction using integers, but some fractions can even contain irrational numbers, (eg: pi/4, sqrt(2)/2; Also: ⅔/3 )-->
##### Appears in these related concepts:
numerator
The number or expression written above the line in a fraction (thus 1 in ½).
##### Appears in these related concepts:
A quadratic polynomial, function or equation.
##### Appears in these related concepts:
A polynomial equation of the second degree.
##### Appears in these related concepts:
rational expression
An expression that can be expressed as the quotient of two polynomials.
##### Appears in these related concepts:
set
A collection of zero or more objects, possibly infinite in size, and disregarding any order or repetition of the objects that may be contained within it.
##### Appears in these related concepts:
variable
A symbol that represents a quantity in a mathematical expression, as used in many sciences