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Rational Expressions

To solve this rational equation, we first find the LCD of the fractions, which is x^2(x-5)(x-6). Then we multiply both sides of the equation by the LCD: x^2(x-5)(x-6) = x^2(x-5)(x-6) Canceling the common factors, we obtain: x^2 = x^2 Since this holds for all real values of x, the solution is all real numbers. Therefore, the solution is x ∈ R.

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RATIONAL EXPRESSIONS
SECTIONS 14.1 – Simplifying Rational Expressions 14.2 – Multiplying and Dividing Rational Expressions 14.3 – Adding and Subtracting Rational Expressions with the Same Denominator and Least Common Denominators 14.4 – Adding and Subtracting Rational Expressions with Different Denominators 14.5 – Solving Equations Containing Rational Expressions 14.6 – Problem Solving with Rational Expressions 14.7 – Simplifying Complex Fractions
SIMPLIFYING RATIONAL EXPRESSIONS
RATIONAL EXPRESSIONS P Rational expressions can be written in the Q form where P and Q are both polynomials and Q 0. Examples of Rational Expressions 3x 2 2 x 4 4x 3y 3x 2 2 2 4x 5 2 x 3xy 4 y 4
Evaluating Rational Expressions To evaluate a rational expression for a particular value(s), substitute the replacement value(s) into the rational expression and simplify the result. Example Evaluate the following expression for y = 2. y 2 2 2 4 4 5 y 5 ( 2) 7 7
Evaluating Rational Expressions In the previous example, what would happen if we tried to evaluate the rational expression for y = 5? y 2 5 2 3 5 y 5 5 0 This expression is undefined!
Undefined Rational Expressions We have to be able to determine when a rational expression is undefined. A rational expression is undefined when the denominator is equal to zero. The numerator being equal to zero is okay (the rational expression simply equals zero).
Undefined Rational Expressions Find any real numbers that make the following rational expression undefined. Example 9 x3 4 x 15 x 45 The expression is undefined when 15x + 45 = 0. So the expression is undefined when x = 3.
Simplifying Rational Expressions Simplifying a rational expression means writing it in lowest terms or simplest form. To do this, we need to use the Fundamental Principle of Rational Expressions If P, Q, and R are polynomials, and Q and R are not 0, PR P QR Q
Simplifying a Rational Expression 1) Completely factor the numerator and denominator. 2) Apply the Fundamental Principle of Rational Expressions to eliminate common factors in the numerator and denominator. Warning! Only common FACTORS can be eliminated from the numerator and denominator. Make sure any expression you eliminate is a factor.
Warm Up Simplify each expression. 1. 2. Factor each expression. 3. x2 + 5x + 6 4. 4x2 – 64 (x + 2)(x + 3) 4(x + 4)(x – 4) 5. 2x2 + 3x + 1 6. 9x2 + 60x + 100 (2x + 1)(x + 1) (3x +10)2
Example 2: Simplifying Rational Expressions Simplify each rational expression, if possible. Identify any excluded values. 4 Factor 14. Divide out common factors. Note that if r = 0, the expression is undefined. Simplify. The excluded value is 0.
Example Simplify the following expression. 7 x 35 7( x 5) 7 2 x 5x x ( x 5) x
Example Simplify the following expression. 2 x 3x 4 ( x 4)( x 1) x 1 2 x x 20 ( x 5)( x 4) x 5
Example Simplify the following expression. 7 y 1( y 7) 1 y 7 y 7
MULTIPLYING AND DIVIDING RATIONAL EXPRESSIONS
MULTIPLYING RATIONAL EXPRESSIONS Multiplying rational expressions when P, Q, R, and S are polynomials with Q 0 and S 0. P R PR Q S QS
Note that after multiplying such expressions, our result may not be in simplified form, so we use the following techniques. Multiplying rational expressions 1) Factor the numerators and denominators. 2) Multiply the numerators and multiply the denominators. 3) Simplify or write the product in lowest terms by applying the fundamental principle to all common factors.
Example Multiply the following rational expressions. 2 6 x 5x 2 3 x x 5 x 1 3 10 x 12 2 5 x x x 2 2 3 4
Example Multiply the following rational expressions. 2 ( m n) m (m n)( m n) m 2 m n m mn ( m n ) m( m n ) m n m n
DIVIDING RATIONAL EXPRESSIONS Dividing rational expressions when P, Q, R, and S are polynomials with Q 0, S 0 and R 0. P R P S PS Q S Q R QR
When dividing rational expressions, first change the division into a multiplication problem, where you use the reciprocal of the divisor as the second factor. Then treat it as a multiplication problem (factor, multiply, simplify).
DIVIDING RATIONAL EXPRESSIONS Example Divide the following rational expression. 2 2 ( x 3) 5 x 15 ( x 3) 25 5 25 5 5 x 15 ( x 3)( x 3) 5 5 x 3 5 5( x 3)
UNITS OF MEASURE Converting Between Units of Measure Use unit fractions (equivalent to 1), but with different measurements in the numerator and denominator. Multiply the unit fractions like rational expressions, canceling common units in the numerators and denominators.
Example Convert 1008 square inches into square feet. 1 ft 1 ft (1008 sq in) 12 in 12 in 1 ft 1 ft (2·2·2·2·3·3·7 in · 2 2 3 in 2 2 3 in in) 7 sq. ft.
A D D I N G AN D S U B T R AC T I N G R AT I ONA L E X P R E S S I ON S W I T H T H E S A M E D E N O M I NATO R AN D L E A S T C O M M ON D E N O M I NATO R S
RATIONAL EXPRESSIONS If P, Q and R are polynomials and Q 0, P Q P Q R R R P Q P Q R R R
ADDING RATIONAL EXPRESSIONS Example Add the following rational expressions. 4p 3 3p 8 4p 3 3p 8 7p 5 2p 7 2p 7 2p 7 2p 7
SUBTRACTING RATIONAL EXPRESSIONS Example Subtract the following rational expressions. 8y 16 8 y 16 8( y 2) 8 y 2 y 2 y 2 y 2
Example Subtract the following rational expressions. 3y 6 3y 6 y 2 3 y 10 y 2 3 y 10 y 2 3 y 10 3( y 2) 3 ( y 5)( y 2) y 5
LEAST COMMON DENOMINATORS To add or subtract rational expressions with unlike denominators, you have to change them to equivalent forms that have the same denominator (a common denominator). This involves finding the least common denominator of the two original rational expressions.
To find a Least Common Denominator: 1) Factor the given denominators. 2) Take the product of all the unique factors. Each factor should be raised to a power equal to the greatest number of times that factor appears in any one of the factored denominators.
Example Find the LCD of the following rational expressions. 1 3x , 6 y 4 y 12 6y 2 3y 2 4 y 12 4( y 3) 2 ( y 3) 2 So the LCD is 2 3 y ( y 3) 12 y ( y 3)
Example Find the LCD of the following rational expressions. 4 4x 2 2 , 2 x 4 x 3 x 10 x 21 x2 4 x 3 ( x 3)( x 1) 2 x 10 x 21 ( x 3)( x 7) So the LCD is (x 3)(x 1)(x 7)
Example Find the LCD of the following rational expressions. 2 3x 4x 2 , 2 5x 5 x 2 x 1 2 2 5 x 5 5( x 1) 5( x 1)( x 1) 2 2 x 2 x 1 ( x 1) 2 So the LCD is 5(x 1)(x - 1)
Example Find the LCD of the following rational expressions. 1 2 , x 3 3 x Both of the denominators are already factored. Since each is the opposite of the other, you can use either x – 3 or 3 – x as the LCD.
MULTIPLYING BY 1 To change rational expressions into equivalent forms, we use the principal that multiplying by 1 (or any form of 1), will give you an equivalent expression. P P P R P R 1 Q Q Q R Q R
EQUIVALENT EXPRESSIONS Example Rewrite the rational expression as an equivalent rational expression with the given denominator. 3 5 9 9y 72 y 4 4 3 3 8y 24 y 9y 5 9 y5 8 y 4 72 y 9
ADDING AND SUBTRACT ING RATIONA L EXPRESSION S WITH DIFFEREN T DENOMINATO RS
UNLIKE DENOMINATORS As stated in the previous section, to add or subtract rational expressions with different denominators, we have to change them to equivalent forms first.
Adding or Subtracting Rational Expressions with Unlike Denominators 1) Find the LCD of all the rational expressions. 2) Rewrite each rational expression as an equivalent one with the LCD as the denominator. 3) Add or subtract numerators and write result over the LCD. 4) Simplify rational expression, if possible.
ADDING WITH UNLIKE DENOMINATORS Example Add the following rational expressions. 15 8 , 7 a 6a 15 8 6 15 7 8 7a 6a 6 7 a 7 6a 90 56 146 73 42a 42a 42a 21a
SUBTRACTING WITH UNLIKE DENOMINATORS Example Subtract the following rational expressions. 5 3 , 2x 6 6 2x 5 3 5 3 2x 6 6 2x 2x 6 2x 6 8 2 2 2 4 2x 6 2( x 3) x 3
Example Subtract the following rational expressions. 7 and 3 2x 3 7 7 3(2 x 3) 3 2x 3 2x 3 2x 3 7 6x 9 7 6x 9 16 6 x 2x 3 2x 3 2x 3 2x 3
ADDING WITH UNLIKE DENOMINATORS Example Add the following rational expressions. 4 x , 2 x2 x 6 x 5x 6 4 x 4 x 2 2 x x 6 x 5 x 6 ( x 3)( x 2) ( x 3)( x 2) 4( x 3) x( x 3) ( x 3)( x 2)( x 3) ( x 3)( x 2)( x 3) 2 4 x 12 x 3 x x 2 x 12 ( x 2)( x 3)( x 3) ( x 2)( x 3)( x 3)
SOLVING EQUATIONS C ONTAINING RATIONAL EXPRESSIONS
SOLVING EQUATIONS First note that an equation contains an equal sign and an expression does not. To solve EQUATIONS containing rational expressions, clear the fractions by multiplying both sides of the equation by the LCD of all the fractions. Then solve as in previous sections. Note: this works for equations only, not simplifying expressions.
Example Solve the following rational equation. 5 7 Check in the original 1 equation. 3x 6 5 7 1 5 7 3 10 6 6x 1 6x 3x 6 5 7 1 10 6x 7x 30 6 10 x 1 7 1 true 6 6
Example Solve the following rational equation. 1 1 1 2 x x 1 3x 2 3x 1 1 1 6x x 1 6x x 1 2x x 1 3x( x 1) 3 x 1 6x 2 3x 3 6x 2 3 3x 2 3x 1 x 1 Continued. 3
Example Continued Substitute the value for x into the original equation, to check the solution. 1 1 1 1 1 2 2 1 3 1 3 1 3 3 3 3 3 3 1 2 4 1 1 3 6 3 3 true 4 4 4 So the solution is x 1 3
Example Solve the following rational equation. x 2 1 1 x 2 7 x 10 3x 6 x 5 x 2 1 1 3x 2 x 5 2 3x 2 x 5 x 7 x 10 3x 6 x 5 3x 2 x 5 3x 2 3x 6 x 5 3x 6 3x x 3x 5 6 6 5x 7 x 7 Continued. 5
Example Continued Substitute the value for x into the original equation, to check the solution. 7 2 5 1 1 2 7 7 7 7 7 10 3 6 5 5 5 5 5 3 5 1 1 49 49 10 21 6 18 25 5 5 5 5 5 5 18 9 18 true So the solution x 7 5 is
Example Solve the following rational equation. 1 2 x 1 x 1 1 2 x 1 x 1 x 1 x 1 x 1 x 1 x 1 2x 1 x 1 2x 2 3 x Continued.
Example Continued Substitute the value for x into the original equation, to check the solution. 1 2 3 1 3 1 1 2 true 2 4 So the solution is x = 3.
Example Solve the following rational equation. 12 3 2 9 a2 3 a 3 a 12 3 2 3 a 3 a 3 a 3 a 9 a2 3 a 3 a 12 3 3 a 23 a 12 9 3a 6 2a 21 3a 6 2a 15 5a Continued. 3 a
Example Continued Substitute the value for x into the original equation, to check the solution. 12 3 2 9 32 3 3 3 3 12 3 2 0 5 0 Since substituting the suggested value of a into the equation produced undefined expressions, the solution is .
Solving Equations with Multiple Variables Solving an Equation With Multiple Variables for One of the Variables 1) Multiply to clear fractions. 2) Use distributive property to remove grouping symbols. 3) Combine like terms to simplify each side. 4) Get all terms containing the specified variable on the same side of the equation, other terms on the opposite side. 5) Isolate the specified variable.
Example Solve the following equation for R1 1 1 1 R R1 R2 1 1 1 RR1 R2 RR1 R2 R R1 R2 R1R2 RR2 RR1 R1R2 RR1 RR2 R1 R2 R RR2 RR2 R1 R2 R
PROBLEM SOLVING WITH RATIONAL EQUATIONS
RATIOS AND RATES Ratio is the quotient of two numbers or two quantities. The ratio of the numbers a and b can also be a written as a:b, or . b The units associated with the ratio are important. The units should match. If the units do not match, it is called a rate, rather than a ratio.
PROPORTIONS Proportion is two ratios (or rates) that are equal to each other. a c b d We can rewrite the proportion by multiplying by the LCD, bd. This simplifies the proportion to ad = bc. This is commonly referred to as the cross produ
SOLVING PROPORTIONS Example Solve the proportion for x. x 1 5 x 2 3 3x 1 5x 2 3x 3 5x 10 2x 7 x 7 2 Continued.
Solving Proportions Example Continued Substitute the value for x into the original equation, to check the solution. 7 1 2 5 7 2 3 2 5 2 5 3 3 true 2 So the solution is x 7 2
Example If a 170-pound person weighs approximately 65 pounds on Mars, how much does a 9000-pound satellite weigh? 170 - pound person on Earth 65 - pound person on Mars 9000 - pound satellite on Earth x - pound satellite on Mars 170x 9000 65 585,000 x 585000 / 170 3441 pounds
SOLVING PROPORTIONS Example Given the following prices charged for various sizes of picante sauce, find the best buy. • 10 ounces for $0.99 • 16 ounces for $1.69 • 30 ounces for $3.29 Continued.
Example Continued Size Price Unit Price 10 ounces $0.99 $0.99/10 = $0.099 16 ounces $1.69 $1.69/16 = $0.105625 30 ounces $3.29 $3.29/30 $0.10967 The 10 ounce size has the lower unit price, so it is the best buy.
FINDING AN UNKNOWN NUMBER Example The quotient of a number and 9 times its reciprocal is 1. Find the number. 1.) Understand Read and reread the problem. If we let n = the number, then 1 = the reciprocal of the number n Continued
Example continued 2.) Translate The quotient is of a number and 9 times its 1 reciprocal 1 n 9 = 1 n Continued
Example continued 3.) 1 Solve n 9 1 n 9 n 1 n n n 1 9 n2 9 n 3, 3 Continued
Example continued 4.) Interpret Check: We substitute the values we found from the equation back into the problem. Note that nothing in the problem indicates that we are restricted to positive values. 1 1 3 9 1 3 9 1 3 3 3 3 1 true 3 3 1 true State: The missing number is 3 or –3.
SOLVING A WORK PROBLEM Example An experienced roofer can roof a house in 26 hours. A beginner needs 39 hours to do the same job. How long will it take if the two roofers work together? 1.) Understand Read and reread the problem. By using the times for each roofer to complete the job alone, we can figure out their corresponding work rates in portion of the job done per hour. Time in hrs Portion job/hr Experienced roofer 26 1/26 Beginner roofer 39 /39 Together t 1/t Continued
Example continued 2.) Translate Since the rate of the two roofers working together would be equal to the sum of the rates of the two roofers working independently, 1 1 1 26 39 t Continued
Example continued 3.) Solve 1 1 1 26 39 t 1 1 1 78t 78t 26 39 t 3t 2t 78 5t 78 t 78 / 5 or 15.6 hours Continued
Example continued 4.) Interpret Check: We substitute the value we found from the proportion calculation back into the problem. 1 1 1 26 39 78 5 3 2 5 true 78 78 78 State: The roofers would take 15.6 hours working together to finish the job.
SOLVING A RATE PROBLEM Example The speed of Lazy River’s current is 5 mph. A boat travels 20 miles downstream in the same time as traveling 10 miles upstream. Find the speed of the boat in still water. 1.) Understand Read and reread the problem. By using the formula d=rt, we can rewrite the formula to find that t = d/r. We note that the rate of the boat downstream would be the rate in still water + the water current and the rate of the boat upstream would be the rate in still water – the water current. Distance rate time = d/r Down 20 r+5 20/(r + 5) Up 10 r–5 10/(r – 5) Continued
Example continued 2.) Translate Since the problem states that the time to travel downstairs was the same as the time to travel upstairs, we get the equation 20 10 r 5 r 5 Continued
Example continued 3.) Solve 20 10 r 5 r 5 20 10 r 5 r 5 r 5 r 5 r 5 r 5 20 r 5 10 r 5 20r 100 10r 50 10r 150 r 15 mph Continued
Example continued 4.) Interpret Check: We substitute the value we found from the proportion calculation back into the problem. 20 10 15 5 15 5 20 10 true 20 10 State: The speed of the boat in still water is 15 mph.
SIMPLIFYING COMPLEX FRACTIONS
COMPLEX RATIONAL FRACTIONS Complex rational expressions (complex fraction) are rational expressions whose numerator, denominator, or both contain one or more rational expressions. There are two methods that can be used when simplifying complex fractions.
SIMPLIFYING COMPLEX FRACTIONS Simplifying a Complex Fraction (Method 1) 1) Simplify the numerator and denominator of the complex fraction so that each is a single fraction. 2) Multiply the numerator of the complex fraction by the reciprocal of the denominator of the complex fraction. 3) Simplify, if possible.
Example x x 4 x 4 2 2 2 2 2 x 4 2 x 4 x x 4 x 4 2 x 4 x 4 2 2 2 2 2
Method 2 for simplifying a complex fraction 1) Find the LCD of all the fractions in both the numerator and the denominator. 2) Multiply both the numerator and the denominator by the LCD. 3) Simplify, if possible.
Example 1 2 2 2 2 y 3 6y 6 4y 1 5 6y2 6y 5y 2 y 6

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Rational Expressions

  • 1.
  • 2.
    SECTIONS 14.1 – SimplifyingRational Expressions 14.2 – Multiplying and Dividing Rational Expressions 14.3 – Adding and Subtracting Rational Expressions with the Same Denominator and Least Common Denominators 14.4 – Adding and Subtracting Rational Expressions with Different Denominators 14.5 – Solving Equations Containing Rational Expressions 14.6 – Problem Solving with Rational Expressions 14.7 – Simplifying Complex Fractions
  • 3.
  • 4.
    RATIONAL EXPRESSIONS P Rational expressions can be written in the Q form where P and Q are both polynomials and Q 0. Examples of Rational Expressions 3x 2 2 x 4 4x 3y 3x 2 2 2 4x 5 2 x 3xy 4 y 4
  • 5.
    Evaluating Rational Expressions Toevaluate a rational expression for a particular value(s), substitute the replacement value(s) into the rational expression and simplify the result. Example Evaluate the following expression for y = 2. y 2 2 2 4 4 5 y 5 ( 2) 7 7
  • 6.
    Evaluating Rational Expressions Inthe previous example, what would happen if we tried to evaluate the rational expression for y = 5? y 2 5 2 3 5 y 5 5 0 This expression is undefined!
  • 7.
    Undefined Rational Expressions Wehave to be able to determine when a rational expression is undefined. A rational expression is undefined when the denominator is equal to zero. The numerator being equal to zero is okay (the rational expression simply equals zero).
  • 8.
    Undefined Rational Expressions Findany real numbers that make the following rational expression undefined. Example 9 x3 4 x 15 x 45 The expression is undefined when 15x + 45 = 0. So the expression is undefined when x = 3.
  • 9.
    Simplifying Rational Expressions Simplifyinga rational expression means writing it in lowest terms or simplest form. To do this, we need to use the Fundamental Principle of Rational Expressions If P, Q, and R are polynomials, and Q and R are not 0, PR P QR Q
  • 10.
    Simplifying a RationalExpression 1) Completely factor the numerator and denominator. 2) Apply the Fundamental Principle of Rational Expressions to eliminate common factors in the numerator and denominator. Warning! Only common FACTORS can be eliminated from the numerator and denominator. Make sure any expression you eliminate is a factor.
  • 11.
    Warm Up Simplify eachexpression. 1. 2. Factor each expression. 3. x2 + 5x + 6 4. 4x2 – 64 (x + 2)(x + 3) 4(x + 4)(x – 4) 5. 2x2 + 3x + 1 6. 9x2 + 60x + 100 (2x + 1)(x + 1) (3x +10)2
  • 12.
    Example 2: SimplifyingRational Expressions Simplify each rational expression, if possible. Identify any excluded values. 4 Factor 14. Divide out common factors. Note that if r = 0, the expression is undefined. Simplify. The excluded value is 0.
  • 13.
    Example Simplify the followingexpression. 7 x 35 7( x 5) 7 2 x 5x x ( x 5) x
  • 14.
    Example Simplify the followingexpression. 2 x 3x 4 ( x 4)( x 1) x 1 2 x x 20 ( x 5)( x 4) x 5
  • 15.
    Example Simplify the followingexpression. 7 y 1( y 7) 1 y 7 y 7
  • 16.
  • 17.
    MULTIPLYING RATIONAL EXPRESSIONS Multiplying rational expressions when P, Q, R, and S are polynomials with Q 0 and S 0. P R PR Q S QS
  • 18.
    Note that aftermultiplying such expressions, our result may not be in simplified form, so we use the following techniques. Multiplying rational expressions 1) Factor the numerators and denominators. 2) Multiply the numerators and multiply the denominators. 3) Simplify or write the product in lowest terms by applying the fundamental principle to all common factors.
  • 19.
    Example Multiply thefollowing rational expressions. 2 6 x 5x 2 3 x x 5 x 1 3 10 x 12 2 5 x x x 2 2 3 4
  • 20.
    Example Multiply thefollowing rational expressions. 2 ( m n) m (m n)( m n) m 2 m n m mn ( m n ) m( m n ) m n m n
  • 21.
    DIVIDING RATIONAL EXPRESSIONS Dividingrational expressions when P, Q, R, and S are polynomials with Q 0, S 0 and R 0. P R P S PS Q S Q R QR
  • 22.
    When dividing rationalexpressions, first change the division into a multiplication problem, where you use the reciprocal of the divisor as the second factor. Then treat it as a multiplication problem (factor, multiply, simplify).
  • 23.
    DIVIDING RATIONAL EXPRESSIONS Example Divide the following rational expression. 2 2 ( x 3) 5 x 15 ( x 3) 25 5 25 5 5 x 15 ( x 3)( x 3) 5 5 x 3 5 5( x 3)
  • 24.
    UNITS OF MEASURE ConvertingBetween Units of Measure Use unit fractions (equivalent to 1), but with different measurements in the numerator and denominator. Multiply the unit fractions like rational expressions, canceling common units in the numerators and denominators.
  • 25.
    Example Convert 1008square inches into square feet. 1 ft 1 ft (1008 sq in) 12 in 12 in 1 ft 1 ft (2·2·2·2·3·3·7 in · 2 2 3 in 2 2 3 in in) 7 sq. ft.
  • 26.
    A D DI N G AN D S U B T R AC T I N G R AT I ONA L E X P R E S S I ON S W I T H T H E S A M E D E N O M I NATO R AN D L E A S T C O M M ON D E N O M I NATO R S
  • 27.
    RATIONAL EXPRESSIONS If P,Q and R are polynomials and Q 0, P Q P Q R R R P Q P Q R R R
  • 28.
    ADDING RATIONAL EXPRESSIONS Example Add the following rational expressions. 4p 3 3p 8 4p 3 3p 8 7p 5 2p 7 2p 7 2p 7 2p 7
  • 29.
    SUBTRACTING RATIONAL EXPRESSIONS Example Subtract the following rational expressions. 8y 16 8 y 16 8( y 2) 8 y 2 y 2 y 2 y 2
  • 30.
    Example Subtract the following rational expressions. 3y 6 3y 6 y 2 3 y 10 y 2 3 y 10 y 2 3 y 10 3( y 2) 3 ( y 5)( y 2) y 5
  • 31.
    LEAST COMMON DENOMINATORS Toadd or subtract rational expressions with unlike denominators, you have to change them to equivalent forms that have the same denominator (a common denominator). This involves finding the least common denominator of the two original rational expressions.
  • 32.
    To find aLeast Common Denominator: 1) Factor the given denominators. 2) Take the product of all the unique factors. Each factor should be raised to a power equal to the greatest number of times that factor appears in any one of the factored denominators.
  • 33.
    Example Find the LCDof the following rational expressions. 1 3x , 6 y 4 y 12 6y 2 3y 2 4 y 12 4( y 3) 2 ( y 3) 2 So the LCD is 2 3 y ( y 3) 12 y ( y 3)
  • 34.
    Example Find the LCDof the following rational expressions. 4 4x 2 2 , 2 x 4 x 3 x 10 x 21 x2 4 x 3 ( x 3)( x 1) 2 x 10 x 21 ( x 3)( x 7) So the LCD is (x 3)(x 1)(x 7)
  • 35.
    Example Find the LCDof the following rational expressions. 2 3x 4x 2 , 2 5x 5 x 2 x 1 2 2 5 x 5 5( x 1) 5( x 1)( x 1) 2 2 x 2 x 1 ( x 1) 2 So the LCD is 5(x 1)(x - 1)
  • 36.
    Example Find the LCDof the following rational expressions. 1 2 , x 3 3 x Both of the denominators are already factored. Since each is the opposite of the other, you can use either x – 3 or 3 – x as the LCD.
  • 37.
    MULTIPLYING BY 1 Tochange rational expressions into equivalent forms, we use the principal that multiplying by 1 (or any form of 1), will give you an equivalent expression. P P P R P R 1 Q Q Q R Q R
  • 38.
    EQUIVALENT EXPRESSIONS Example Rewrite therational expression as an equivalent rational expression with the given denominator. 3 5 9 9y 72 y 4 4 3 3 8y 24 y 9y 5 9 y5 8 y 4 72 y 9
  • 39.
    ADDING AND SUBTRACTING RATIONA L EXPRESSION S WITH DIFFEREN T DENOMINATO RS
  • 40.
    UNLIKE DENOMINATORS As statedin the previous section, to add or subtract rational expressions with different denominators, we have to change them to equivalent forms first.
  • 41.
    Adding or SubtractingRational Expressions with Unlike Denominators 1) Find the LCD of all the rational expressions. 2) Rewrite each rational expression as an equivalent one with the LCD as the denominator. 3) Add or subtract numerators and write result over the LCD. 4) Simplify rational expression, if possible.
  • 42.
    ADDING WITH UNLIKEDENOMINATORS Example Add the following rational expressions. 15 8 , 7 a 6a 15 8 6 15 7 8 7a 6a 6 7 a 7 6a 90 56 146 73 42a 42a 42a 21a
  • 43.
    SUBTRACTING WITH UNLIKEDENOMINATORS Example Subtract the following rational expressions. 5 3 , 2x 6 6 2x 5 3 5 3 2x 6 6 2x 2x 6 2x 6 8 2 2 2 4 2x 6 2( x 3) x 3
  • 44.
    Example Subtract thefollowing rational expressions. 7 and 3 2x 3 7 7 3(2 x 3) 3 2x 3 2x 3 2x 3 7 6x 9 7 6x 9 16 6 x 2x 3 2x 3 2x 3 2x 3
  • 45.
    ADDING WITH UNLIKEDENOMINATORS Example Add the following rational expressions. 4 x , 2 x2 x 6 x 5x 6 4 x 4 x 2 2 x x 6 x 5 x 6 ( x 3)( x 2) ( x 3)( x 2) 4( x 3) x( x 3) ( x 3)( x 2)( x 3) ( x 3)( x 2)( x 3) 2 4 x 12 x 3 x x 2 x 12 ( x 2)( x 3)( x 3) ( x 2)( x 3)( x 3)
  • 46.
    SOLVING EQUATIONS C ONTAININGRATIONAL EXPRESSIONS
  • 47.
    SOLVING EQUATIONS First notethat an equation contains an equal sign and an expression does not. To solve EQUATIONS containing rational expressions, clear the fractions by multiplying both sides of the equation by the LCD of all the fractions. Then solve as in previous sections. Note: this works for equations only, not simplifying expressions.
  • 48.
    Example Solve the following rational equation. 5 7 Check in the original 1 equation. 3x 6 5 7 1 5 7 3 10 6 6x 1 6x 3x 6 5 7 1 10 6x 7x 30 6 10 x 1 7 1 true 6 6
  • 49.
    Example Solve thefollowing rational equation. 1 1 1 2 x x 1 3x 2 3x 1 1 1 6x x 1 6x x 1 2x x 1 3x( x 1) 3 x 1 6x 2 3x 3 6x 2 3 3x 2 3x 1 x 1 Continued. 3
  • 50.
    Example Continued Substitutethe value for x into the original equation, to check the solution. 1 1 1 1 1 2 2 1 3 1 3 1 3 3 3 3 3 3 1 2 4 1 1 3 6 3 3 true 4 4 4 So the solution is x 1 3
  • 51.
    Example Solve thefollowing rational equation. x 2 1 1 x 2 7 x 10 3x 6 x 5 x 2 1 1 3x 2 x 5 2 3x 2 x 5 x 7 x 10 3x 6 x 5 3x 2 x 5 3x 2 3x 6 x 5 3x 6 3x x 3x 5 6 6 5x 7 x 7 Continued. 5
  • 52.
    Example Continued Substitutethe value for x into the original equation, to check the solution. 7 2 5 1 1 2 7 7 7 7 7 10 3 6 5 5 5 5 5 3 5 1 1 49 49 10 21 6 18 25 5 5 5 5 5 5 18 9 18 true So the solution x 7 5 is
  • 53.
    Example Solve thefollowing rational equation. 1 2 x 1 x 1 1 2 x 1 x 1 x 1 x 1 x 1 x 1 x 1 2x 1 x 1 2x 2 3 x Continued.
  • 54.
    Example Continued Substitutethe value for x into the original equation, to check the solution. 1 2 3 1 3 1 1 2 true 2 4 So the solution is x = 3.
  • 55.
    Example Solve the followingrational equation. 12 3 2 9 a2 3 a 3 a 12 3 2 3 a 3 a 3 a 3 a 9 a2 3 a 3 a 12 3 3 a 23 a 12 9 3a 6 2a 21 3a 6 2a 15 5a Continued. 3 a
  • 56.
    Example Continued Substitutethe value for x into the original equation, to check the solution. 12 3 2 9 32 3 3 3 3 12 3 2 0 5 0 Since substituting the suggested value of a into the equation produced undefined expressions, the solution is .
  • 57.
    Solving Equations withMultiple Variables Solving an Equation With Multiple Variables for One of the Variables 1) Multiply to clear fractions. 2) Use distributive property to remove grouping symbols. 3) Combine like terms to simplify each side. 4) Get all terms containing the specified variable on the same side of the equation, other terms on the opposite side. 5) Isolate the specified variable.
  • 58.
    Example Solve thefollowing equation for R1 1 1 1 R R1 R2 1 1 1 RR1 R2 RR1 R2 R R1 R2 R1R2 RR2 RR1 R1R2 RR1 RR2 R1 R2 R RR2 RR2 R1 R2 R
  • 59.
  • 60.
    RATIOS AND RATES Ratiois the quotient of two numbers or two quantities. The ratio of the numbers a and b can also be a written as a:b, or . b The units associated with the ratio are important. The units should match. If the units do not match, it is called a rate, rather than a ratio.
  • 61.
    PROPORTIONS Proportion is tworatios (or rates) that are equal to each other. a c b d We can rewrite the proportion by multiplying by the LCD, bd. This simplifies the proportion to ad = bc. This is commonly referred to as the cross produ
  • 62.
    SOLVING PROPORTIONS Example Solvethe proportion for x. x 1 5 x 2 3 3x 1 5x 2 3x 3 5x 10 2x 7 x 7 2 Continued.
  • 63.
    Solving Proportions Example Continued Substitute the value for x into the original equation, to check the solution. 7 1 2 5 7 2 3 2 5 2 5 3 3 true 2 So the solution is x 7 2
  • 64.
    Example If a 170-poundperson weighs approximately 65 pounds on Mars, how much does a 9000-pound satellite weigh? 170 - pound person on Earth 65 - pound person on Mars 9000 - pound satellite on Earth x - pound satellite on Mars 170x 9000 65 585,000 x 585000 / 170 3441 pounds
  • 65.
    SOLVING PROPORTIONS Example Giventhe following prices charged for various sizes of picante sauce, find the best buy. • 10 ounces for $0.99 • 16 ounces for $1.69 • 30 ounces for $3.29 Continued.
  • 66.
    Example Continued Size Price Unit Price 10 ounces $0.99 $0.99/10 = $0.099 16 ounces $1.69 $1.69/16 = $0.105625 30 ounces $3.29 $3.29/30 $0.10967 The 10 ounce size has the lower unit price, so it is the best buy.
  • 67.
    FINDING AN UNKNOWNNUMBER Example The quotient of a number and 9 times its reciprocal is 1. Find the number. 1.) Understand Read and reread the problem. If we let n = the number, then 1 = the reciprocal of the number n Continued
  • 68.
    Example continued 2.) Translate The quotient is of a number and 9 times its 1 reciprocal 1 n 9 = 1 n Continued
  • 69.
    Example continued 3.) 1 Solve n 9 1 n 9 n 1 n n n 1 9 n2 9 n 3, 3 Continued
  • 70.
    Example continued 4.) Interpret Check: We substitute the values we found from the equation back into the problem. Note that nothing in the problem indicates that we are restricted to positive values. 1 1 3 9 1 3 9 1 3 3 3 3 1 true 3 3 1 true State: The missing number is 3 or –3.
  • 71.
    SOLVING A WORKPROBLEM Example An experienced roofer can roof a house in 26 hours. A beginner needs 39 hours to do the same job. How long will it take if the two roofers work together? 1.) Understand Read and reread the problem. By using the times for each roofer to complete the job alone, we can figure out their corresponding work rates in portion of the job done per hour. Time in hrs Portion job/hr Experienced roofer 26 1/26 Beginner roofer 39 /39 Together t 1/t Continued
  • 72.
    Example continued 2.) Translate Since the rate of the two roofers working together would be equal to the sum of the rates of the two roofers working independently, 1 1 1 26 39 t Continued
  • 73.
    Example continued 3.) Solve 1 1 1 26 39 t 1 1 1 78t 78t 26 39 t 3t 2t 78 5t 78 t 78 / 5 or 15.6 hours Continued
  • 74.
    Example continued 4.) Interpret Check: We substitute the value we found from the proportion calculation back into the problem. 1 1 1 26 39 78 5 3 2 5 true 78 78 78 State: The roofers would take 15.6 hours working together to finish the job.
  • 75.
    SOLVING A RATEPROBLEM Example The speed of Lazy River’s current is 5 mph. A boat travels 20 miles downstream in the same time as traveling 10 miles upstream. Find the speed of the boat in still water. 1.) Understand Read and reread the problem. By using the formula d=rt, we can rewrite the formula to find that t = d/r. We note that the rate of the boat downstream would be the rate in still water + the water current and the rate of the boat upstream would be the rate in still water – the water current. Distance rate time = d/r Down 20 r+5 20/(r + 5) Up 10 r–5 10/(r – 5) Continued
  • 76.
    Example continued 2.) Translate Since the problem states that the time to travel downstairs was the same as the time to travel upstairs, we get the equation 20 10 r 5 r 5 Continued
  • 77.
    Example continued 3.) Solve 20 10 r 5 r 5 20 10 r 5 r 5 r 5 r 5 r 5 r 5 20 r 5 10 r 5 20r 100 10r 50 10r 150 r 15 mph Continued
  • 78.
    Example continued 4.) Interpret Check: We substitute the value we found from the proportion calculation back into the problem. 20 10 15 5 15 5 20 10 true 20 10 State: The speed of the boat in still water is 15 mph.
  • 79.
  • 80.
    COMPLEX RATIONAL FRACTIONS Complexrational expressions (complex fraction) are rational expressions whose numerator, denominator, or both contain one or more rational expressions. There are two methods that can be used when simplifying complex fractions.
  • 81.
    SIMPLIFYING COMPLEX FRACTIONS Simplifyinga Complex Fraction (Method 1) 1) Simplify the numerator and denominator of the complex fraction so that each is a single fraction. 2) Multiply the numerator of the complex fraction by the reciprocal of the denominator of the complex fraction. 3) Simplify, if possible.
  • 82.
    Example x x 4 x 4 2 2 2 2 2 x 4 2 x 4 x x 4 x 4 2 x 4 x 4 2 2 2 2 2
  • 83.
    Method 2 forsimplifying a complex fraction 1) Find the LCD of all the fractions in both the numerator and the denominator. 2) Multiply both the numerator and the denominator by the LCD. 3) Simplify, if possible.
  • 84.
    Example 1 2 2 2 2 y 3 6y 6 4y 1 5 6y2 6y 5y 2 y 6