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6.1 The Fundamental Property of Rational Expressions A rational expression (also called an algebraic fraction) is an expression of
the form Objective 1: To evaluate an algebraic fraction, simply substitute the value of the variable into the expression and simplify.  when x = 3.
Solution:
ASSIGNMENT: PAGE 385, problems 3, 5, 7 Objective 2: The numerator of a fraction may be any real number (remember that zero is a real number). The denominator of a fraction may NOT be zero – if the denominator is zero then the fraction is not defined. (A fraction is a way of expressing division and division by zero is not defined.) Find the values for which a rational expression is undefined by setting the denominator equal to zero and solving this equation. The solution(s) of the equation is the value for which the expression is not defined. Problem: Find all values for which the rational expression is undefined.
So, the expression is not defined for values of r = -4 and r = -2. ASSIGNMENT: PAGE 385, problems 13 – 23, odd Objective 3: A fraction is in lowest terms when the numerator and denominator have no common factors. The Fundamental Property of Fractions states that both the numerator and the denominator of any fraction may be multiplied or divided by the same nonzero number without altering the value of the fraction. This is what allows us to build or reduce fractions. To reduce a fraction: Factor both the numerator and the denominator of the fraction completely. Cancel any factors common to both numerator and denominator. The parts remaining give the factored form of the fraction in lowest terms. ***YOU MAY NEVER CANCEL TERMS! YOU MAY ONLY CANCEL FACTORS! YOU MUST FIRST FACTOR AND THEN CANCEL!**** Problem: Reduce to lowest terms:A. Solution: A. B. 
C. When working with polynomials, remember that expressions like a – 6 and 6
– a are opposites. Problem: Reduce to lowest terms:A. Solution: A.
B. ASSIGNMENT: PAGE 386, problems 27 – 51, every other odd problems 53, 57, 59 problems 63, - 69, odd
Objective 4: When a fraction is negative, the negative sign may appear in any one of three places. 1. In front of the fraction: (A fraction bar is a symbol of grouping.)
2. In the numerator: (be sure to use parenthesis or simplify!)
3. In the denominator: (be sure to use parenthesis or simplify!)
Solution:
ASSIGNMENT: PAGE 387, problems 73, 75
6.2 Multiplying and Dividing Rational Expressions Objective 1: To multiply rational expressions, simply multiply the numerators together and multiply the denominators together and reduce this new fraction.
Problems: Multiply and write each answer in lowest terms. A.
B. NOTE: It is also possible to cancel before multiplying. If all of the factors common to the numerator and the denominator are cancelled then the product of the remaining factor is the reduced form. NOTE: You must first factor and then cancel. ASSIGNMENT: PAGE 392, problems 5, 9, 13 Objective 2: To divide fractions, we must multiply by the reciprocal of the divisor (the second fraction).
B.
C. ASSIGNMENT: PAGE 392-3, problems 17, 21, 25 problems 29 – 55, odd
6.4 Adding and Subtracting Rational Fractions Objective 1: To add fractions having the same denominators, add the numerators together and write this sum over the denominator. Then reduce the fraction.
B. Objective 2: To add fractions with different denominators, follow these steps. 1. Find the least common denominator (LCD) of the fractions. 2. Rewrite each fraction as an equivalent fraction with this new denominator. 3. Add the numerators and write the sum over the LCD. 4. Reduce. REMINDER: To find the least common denominator (LCD): ***Factor each denominator completely. ***List each different factor used the greatest number of times that it appears in any of the denominators. ***This product is the LCD. A. B. C.
D.
Objective 3: To subtract fractions having the same denominator, subtract the numerators and write this difference over the denominator. Reduce the fraction to lowest terms.
To subtract fractions with different denominators, follow these steps: 1. Find the least common denominator (LCD) of the fractions. 2. Rewrite each fraction as an equivalent fraction with this new denominator. 3. Subtract the numerators and write the sum over the LCD. 4. Reduce. ***REMEMBER to be very careful when subtracting expressions! Expressions following a subtraction sign must be enclosed in parentheses because each term of the expression is being subtracted. Problems: Subtract and write each answer in lowest terms. A.
B.
ASSIGNMENT: PAGE 407-8, problems 11, 15, 19, 25 – 47 (odd) 51, 57, 61, 63, 67
6.5 Complex Fractions A complex fraction is a fraction that contains a fraction, mixed number or the decimal form of a number in either the numerator, denominator, or both. Examples of complex fractions: There are two common ways to simplify a complex fraction. Method 1: Simplify the numerator of the complex fraction. Simplify the denominator of the complex fraction. Write the complex fraction as a division problem. Perform the division and make sure your answer is in lowest terms. Problem: Simplify the complex fraction
Solution (using Method 1):
Simplify the numerator.
Simplify the denominator. The division problem:
Problem: Simplify the complex fraction
Solution (using Method 1):
Simplify the numerator.
Simplify the denominator. The division problem: 
Solution (using Method 1):
Simplify the numerator.
Simplify the denominator
The division problem:
Method 2: Find the LCM of the complex fraction. Multiply both the numerator and the denominator of the complex fraction by the LCM. Simplify. Problem: Simplify the complex fraction
Solution (using Method 2):
The LCM is 60.
Multiply by the LCM. ASSIGNMENT: PAGE 415 - 416, problems 7 - 35 odd
6.6 Solving Equations with Rational Expressions Objective 1: It is very important to be able to distinguish between expressions, which are composed of sums and differences, and equations, which are statements of equality of expressions. Examples: is an equation.
Objective 2: When solving equations involving algebraic fractions, consider eliminating the fractions by multiplying each term of the equation by the LCD. The resulting equation is an equivalent equation to the original and thus has the same solution. The solutions to algebraic fractions must always be checked to ensure that the solution does not yield a zero in the denominator. If it does, then that solution is discarded. Problem: Solve for x:
Solution:
LCD is 15. 
Check:
Solution:
LCD is x + 1.
Check:
The only answer obtained yields a zero in the denominator so the original equation has no solution. Problem: Solve for p: 
Solution:
LCD is p(p – 2)(p – 1). 
Check:
True so the solution is correct. NOTE: When solving a rational equation that consists only of a fraction on the left side of the equation and only a fraction on the right side of the equation, you may set the cross-products equal as a "short cut" for multiplying by the LCD. 
Solution:
(Both of these answers will check.) Objective 3: When solving a formula for a specific variable, the steps mirror those used to solve equations. Treat the variable that you are solving for as the only variable and treat the other variables as constants. Problem: Solve for y:
Solution: 
Problem:
Solve for a:
Solution:
ASSIGNMENT: PAGE 423-5 problems 17 – 69 odd, 73 – 85 odd |
, where P and Q are
polynomials and Q cannot be equal to 0 (zero).
B.
C.
B.
is an expression (there is no equal sign!)
is an expression (there is no equal sign!)
is an equation.
True The solution is correct.
