Solution of Rational Equations.

By working on this lesson you will learn how to simplify rational equations, how to get the Least Common Denominator, and to solve rational equations.

Lesson's Content

Lesson's Glossary

Equation: A mathematical statement that has to expressions joined by the = sign. It has the right side of the equation (expression1) and the left side of the equation (expression 2)

Equivalent equations: An equation that when solved have the same solution set over a given domain.

Equivalent expressions:  All expressions that represent the same number for any value of the variable that they contain.

Extraneous Solution: After an equation is solved, sometimes the number yielded in the solution doesn’t work for the original equation. This is an extraneous solution. Rational, radical, logarigthmic and absolute value equations need to be verified for this kind of solutions.

LCM: The least common multiple (LCM) of two numbers is the smallest number (different to zero) which is a multiple of both (all).

Rational expression: An expression with fractions, especially when there is a variable in the denominator. The variable can not make the denominator equal to zero

Rational function: A rational function is a function of the form f(x) = P(x)/Q(x), where P(x) and Q(x) are polynomial functions and Q(x) different than 0.

Sides of an equation:  The two expressions at both sides of the equal sign in an equation. Left and right.

Sides of an inequality: The two expressions at both sides of the inequality sign in an inequality. Left and right.

Solve an equation:  To find the solution, to find the answer, to get to know the value for which a variable stands for.

Term: A form of grouping one or more numerical and/or variable factors by means of multiplication and division. 

Variable: A letter used to represent a number. When the variable is part of an equation, it is possible to find the value for which it stands for by solving the equation. This is the solution (s) of the equation.

Variable expression: Any expression containing one or more variables.

Interactive Algebraic Applet: Rational Functions

This interactive will help you to explore the graph for a rational equation formed by two quadratic equations in the numberator, and denominator of the rational expression.

Something to take into account is:

Domain
The domain is valid for all real values except where the denominator, Q(x) = 0.

x-intercepts
The roots, zeros, solutions, x-intercepts of the rational function are in places where P(x) = 0.

Vertical Asymptotes
Vertical asymptotes are located in the x-values that make the denominator zero. The asymptotes may go up, or down; and they may go in the same direction at the left, and right of the asymptote, or in different directions (one up, the other down). This depends in whether you have an even or odd number of factors.

Horizontal Asymptotes
A horizontal asymptote goes far right, and/or far left of the graph. They are not asymptotic in the middle.
To find the location of the horizontal asymptote you look at the degrees of the numerator (n) and denominator (m).

• If n<m, the x-axis, y=0 is the horizontal asymptote.
• If n=m, then the ratio of the leading coefficients determines the y value for the horizontal asymptote: Top Leading Coefficient/Bottom Leading Coefficient.
• If n>m, there is not horizontal asymptote, and  if n=m+1, there is an oblique or slant asymptote.

Holes
In some rational functions, a factor may appear in the numerator and in the denominator. Assuming the factor (x-a) is in the numerator and denominator. Because the factor is in the denominator, x=a won’t be part of the domain of the function. The outcome for this may be: There will either be a vertical asymptote at x=k, or there will be a hole at x=k.

Cases:

• We have more (x-a) factors in the denominator. After we reduce the rational expression, the (x-a) is still in the denominator. Factors in the denominator yield vertical asymptotes. So, we have a vertical asymptote at x=a.
• We have more (x-a) factors in the numerator. After we reduce the rational expression, the (x-a) is still in the numerator. Factors in the numerator yield x-intercepts. Nevertheless, because we can’t have x=a, there is a hole in the graph on the x-axis.
• We have equal numbers of (x-a) factors in both the numerator and denominator. After we reduce the rational expression, there is no (x-a) left in either, numerator or denominator. No (x-a) in the denominator, then  no vertical asymptote at x=a. No  (x-a) in the numerator then  no x-intercept at x=a. There is just a hole in the graph, in a different place than the x-axis.

Observe the picture below in reference to the applet.

Oblique Asymptotes
If the degree of the numerator is exactly one more than the degree of the denominator, then the graph of the rational function will have an oblique asymptote.
To determine the equation of the oblique asymptote, perform long division by dividing the denominator into the numerator, or synthetic division; when this works. To find the equation of the oblique asymptote, discard the remainder after you performed long division.

The applet shown below, highlights the oblique or slant asymptote. Just set the value for the exponent in the leading coefficient of the numerator

to 3.

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