Elementary Algebra Chapter 6
RATIONAL EXPRESSIONS AND EQUATIONS

Before beginning this chapter, please review Section 1.2.

Rational numbers (common fractions) are ratios of integers.
Rational expressions are ratios of polynomials.
The rules for operating (adding, subtracting, multiplying, and dividing) with rational expressions are the same as operating with rational numbers (common fractions).

From Section 6.1
Definition:
A rational expression is a quotient of two polynomials.

Fundamental Property of Fractions: If a represents a real number and b and c represent non-zero real numbers then
                                                                              

FACT: The Fundamental Property of Fractions is as valid for rational expressions as it is for common fractions. The only difference is that when applying the property to rational expressions one must interpret a, b and c as polynomials

From Section 6.2
Rule for Multiplying Fractions:
If a, b, c, and d represent real numbers and b isn not 0 and d is not 0 then
                                                            

FACT: The Rule for Multiplying Fractions is as valid for rational expressions as it is for common fractions. The only difference is that when applying the rule to rational expressions one must interpret a, b,c and d as polynomials.
PROCESS:
A good way to remember this rule is to observe that it states:
                        The numerator of a product is the product of the numerators and
                         the denominator of a product is the product of the denominators.

Division of Fractions: If a represents a real number and b, c, and d represent non-zero real numbers, then
                                                                          
FACT: The definition of division as applied to fractions is as valid for rational expressions as it is for common fractions. The only difference is that when applying the rule to rational expressions one must interpret a, b,c and d as polynomials.


An alternate depiction of this definition of division whether applied to fractions or rational expressions is the familiar

                                                                       
This is of course an adaptation (to fraction notation) of the normal definition of division as shown here.
                                                                          

From Section 6.3 and Section 6.4
Adding Fractions with the Same Denominator: If a, b, and d represent real numbers and d is not 0, then
                                                                         
FACT: Adding fractions with the same denominator is as valid for rational expressions as it is for common fractions. The only difference is that when applying the rule to rational expressions one must interpret a, b and d as polynomials.

Adding Fractions with the Different Denominators: If a, b, c, and d represent real numbers and and b is not 0 and d is not 0, then the sum is computed by expanding the two fractions to fractions with the same denominator ( some common multiple of b and d) and then adding the two fractions with the same denominator.

FACT: Adding fractions with the different denominators is as valid for rational expressions as it is for common fractions. The only difference is that when applying the rule to rational expressions one must interpret a, b, c, and d as polynomials.

Subtraction of Fractions: Subtraction of fractions is defined as an addition as shown here.
                                                     
This is of course an adaptation (to fraction notation) of the normal definition of subtraction as shown here.
                                                       
PROCESS: To subtract one fraction or rational expression from another, change the problem to an addition problem and proceed according to the rules for addition.

 

From Section 6.5

PROCESS: To simplify a complex fraction simply perform the indicated division.

Note: If any of the numerators or denominators of the fractions involved contain a sum or difference, then the indicated addition (or subtraction) should be performed before attempting the division.

Note: Recall that all divisions are performed by converting to multiplication.


From Section 6.6

Definition: A rational equation is an equation which contains one or more rational expressions.

RECALL: If any expression is added to both sides of an equation the resulting equation is equivalent to the original equation.

RECALL: If both sides of an equation are multiplied by the same non-zero real number, the resulting equation is equivalent to the original equation.

FACT -- Multiplying by a Variable Expression: When both sides of an equation are multiplied by an expression which contains a varible, the resulting equation is generally NOT equivalent to the original equation.

FACT -- Multiplying by a Variable Expression: When both sides of an equation are multiplied by an expression which contains a varible, the solution set of the resulting equation CONTAINS the solution set of the original equation.

PROCESS: To solve a rational equation;
1) Multiply both sides of the equation by the LCD of all rational expressions in the equation.
2) Solve the resulting equation to obtain possible solutions of the original equation.
3) Determine the solution set of the original equation by checking each of the possible solutions in the original equation.

FACT: If both sides of a rational equation are multiplied by the LCD of the rational expressions in the equation and if that LCD is a number (no variables), then the resulting equation is equivalent to the original equation.

FACT: The only way a possible rational solution (as found by the above process) can fail to be a solution of the original equation is for the possible solution to cause a denominator in the original equation to be zero . Consequently, possible solutions can be checked by simply insuring that they do not cause any denominator to be zero.


Zero Quotient Property: A fraction is equal to zero if and only if its numerator is zero.

ALTERNATE PROCESS: To solve a rational equation;
1) Add or subtract appropriate rational expressions to both sides of the equation to obtain 0 on one side of the equation.
2) Perform all the indicated additions and subtractions of rational expressions.
3) Use the Zero Factor Property to conclude the numerator must be zero.
4) Solve the resulting equation to obtain possible solutions of the original equation.
5) Determine the solution set of the original equation by checking each of the possible solutions in the original equation.