RATIONAL EXPRESSIONS AND EQUATIONS Chapter 4
4.1 EQUIVALENT RATIONAL EXPRESSIONS Chapter 4
RATIONAL EXPRESSIONS What is a rational number? A Rational Number is the ratio of two integers Examples: 2 3 7 12 5 312.
WHAT MIGHT A RATIONAL EXPRESSION BE? A rational expression is an algebraic fraction (includes variables) with a numerator and a denominator that are polynomials. It is any expression that can be written as the quotient of two polynomials, in the form Px ( ) where Qx ( ) 0. Qx ( ) Examples: 1 x m m + 1 y 2-1 y 2 + 2y + 1 x 2-4
ARE THE FOLLOWING RATIONAL EXPRESSIONS? WHY OR WHY NOT? 4 5 2 2x x 4 y x 1 5 2 x x 2y 4 2
NON-PERMISSIBLE VALUES What value can x not have? 1 x For all rational expressions with variables in the denominator, we need to define the non-permissible values. These are the values for a variable that makes an expression undefined. In a rational expression, this is a value that results in a denominator of zero. These non-permissible values are the restrictions on the domain of the rational expression.
DETERMINE THE NON-PERMISSIBLE VALUES FOR FOLLOWING: A) 5 4x 2 x 2x B) C) 4 2 x 4 x 2x D) x 2 x x 7 ( x 1)( x 2) E) F) x 2 4 9 2 x 1 x 1
PRACTICE: WHAT ARE THE NON-PERMISSIBLE VALUES? A. 5x 2 B. 2 25x 16 5x 2 2 2x 8
5x 2 C. D. 2 20x 15x 5x 2 x 2 25
WRITE A RATIONAL EXPRESSION THAT HAS NON-PERMISSIBLE VALUES OF: A) 2 and 4 B) 0, -2, and 3
WRITE A RATIONAL EXPRESSION THAT HAS NON-PERMISSIBLE VALUES OF: 1 4 C) D) 3 and 4 7
NON-PERMISSIBLE VS. INADMISSIBLE VALUES Non-Permissible Values: values of a variable that make the denominator of a rational expression equal zero. Inadmissible Values: values of a variable that do not make sense in the context of a given problem.
EXAMPLE: IF A BOAT TRAVELLED 20 KM WITH A SPEED OF X KM/H, THE TIME TAKEN FOR THE TRIP WOULD BE 20 REPRESENTED BY THE EXPRESSION x. x 0 is a non-permissible value since it makes the denominator = 0. Negative x-values are inadmissible since they result in negative time (which doesn t make sense!)
INDEPENDENT PRACTICE Pages 223-224 #3(find non-permissible values only), 9a (i, ii,),10, 11a (i, ii,), 16
EQUIVALENT FRACTIONS
8 12 WRITING EQUIVALENT FRACTIONS INVOLVING ONLY NUMBERS: (RATIONAL NUMBERS) To find equivalent rational numbers (fractions), we multiply or divide both the numerator and denominator by the same number. Example: Write two rational numbers that are equivalent to: 8 A) 12 25 B) 100
EQUIVALENT RATIONAL EXPRESSIONS (FRACTIONS WITH VARIABLES) To find equivalent rational expressions, we Multiply or divide both the numerator and denominator by the same number. EX. x x 3 2 Do not multiply by a factor that introduces a new restriction. That is, do not multiply by something like: EX. What happens to the restrictions when x 3 is multiplied by x x 2 x x x x 1 or x 1
Equivalent rational expressions must have the same restrictions. However we can divide the numerator and denominator by a common factor provided we keep the same restrictions. Ex. Consider: x x 1 2 x 2x A) What are the restrictions (non-permissible values)? B) Determine the common factor and divide numerator and denominator. State the restrictions.
EXAMPLES: 1. FOR EACH EXPRESSION, STATE THE RESTRICTION(S) AND WRITE T WO EQUIVALENT RATIONAL EXPRESSIONS. A) 4 x
B) Write a rational expression that is equivalent to 4x2 +8x 4x Firstly, what are the nonpermissible values? 4x 2 +8x, x ¹ 0 4x Can I possibly factor either the numerator or the denominator? 4x 2 +8x 4x = 4x(x + 2) 4x We need to write our non-permissible values at the end. = x + 2, x ¹ 0
WRITE A SECOND RATIONAL EXPRESSION THAT IS EQUIVALENT TO 4x2 +8x 4x What happens when we multiply by equivalent expression? x x? Is the new equation an Yes, the new expression is an equivalent expression since both expressions have the same restrictions.
C) 6n 3 3 2n 1
EXAMPLE 2 For each of the following, determine if the rational expressions are equivalent. A) 9 3x -1 and -18 2-6x a) What would I need to multiply 9 by to get 18? 9 9 2 3x 1 3x 1 2 18 6x 2 18 2 6x Two conditions in order for the expressions to be equivalent: Same restrictions If you multiply or divide by a common factor, you should get the other expression The expressions are equivalent!
B) 2 2x 4x and x 1 2x
B) 2 2x 4x and x 1 2x Another method is to check using substitution. So, choose a value that you d like to put in for x. Let s try x = 3. 2 2(3) 1 4(3) 3 3 1 1 2(3) 3 The expressions are not equal for x = 3, so they aren t equivalent.
EXAMPLE 3
SUMMARY To determine whether two rational expressions are equivalent: The numerator and denominator of one expression must be multiplied or divided by the same value to produce the other expression The two expressions must have the same restrictions (same non-permissible values).
SUBSTITUTION We can use substitution to show that two rational expressions are NOT equivalent. If we can come up with at least one value for a variable that when substituted into two rational expressions makes them work out to be different values, then the expressions are NOT equal. However, substitution CANNOT be used to show that two rational expressions ARE equal. Expressions may be equal for some values but not necessarily all values.
EXAMPLES A)Explain why 3 and 3( x 1) are NOT equivalent expressions. 2x 2x( x 1)
B)Using Substitution, explain why 5x 4 and 3x 2 are NOT equivalent expressions. 2x 2x (Use and ) x 1 x 2
PAGES 223-224 #1, 3, 5, 6, 14, 15, Independent Practice
4.2 SIMPLIFYING RATIONAL EXPRESSIONS Chapter 4
SIMPLIFYING RATIONAL EXPRESSIONS IS SIMILAR TO SIMPLIFYING ALGEBRAIC FRACTIONS Example: Simplify each fraction. A. 3 B. 4 C. 6 10 25 15
SIMPLIFYING RATIONAL EXPRESSIONS RESULTS IN AN EXPRESSION THAT IS EASIER TO WORK WITH THAN THE ORIGINAL Simplify the following rational expression: What are both 24 and 18 divisible by? 6 What are the nonpermissible values?
EXAMPLE Simplify the following rational expression: Can I factor the numerator? What are the non-permissible values? Can 5x be factored out of 15x 3? Don t forget your nonpermissible values at the end.
STEPS FOR SIMPLIFYING RATIONAL EXPRESSIONS 1. Find the non-permissible values. 2. Completely factor the numerator and denominator. 3. Cancel any factors that are common to both. 4. Write the simplified expression and be sure to state the restrictions for the expression.
EXAMPLE Simplify the following rational expression: Non-permissible values: 3m 3 4m 2 = 0 m 2 (3m 4) = 0 m 2 = 0 3m 4 = 0 m = 0 3m = 4 m = 4/3 So, m 0, 4/3
SIMPLIFY THE FOLLOWING AND STATE ANY RESTRICTIONS. B. 6x 3 8x 4
C. 6x 27x 2 4x 18x 3 2
2 D. x 9 2 2x 6x
2 E. 3x 12 6x 12
4 F. 5x 405 10x 30
G. 16x 12x 3 32x 18x 3 2
2 H. 4x 6x 4 16x 81
IDENTIFY AND CORRECT ANY ERRORS IN THE FOLLOWING SIMPLIFICATIONS: A. = =
B. =
C.
PG. 229-231, #3, 4, 5, 8, 9, 10, 13. Independent Practice
4.3 MULTIPLYING AND DIVIDING RATIONAL Chapter 4 EXPRESSIONS
STEPS FOR MULTIPLYING RATIONAL EXPRESSIONS 1. Factor the numerators and denominators of each expression. 2. Find the non-permissible values. Look at ALL of the denominators. 3. Multiply the numerators and the denominators together. Cancel common factors. 4. Write the simplified expression including the nonpermissible values.
Simplify each of the following products: a) Step 1: Factor the numerators and denominators of each expression. Step 3: Multiply the numerators and the denominators together. Cancel common factors. Step 2: Find the non-permissible values. Look at all of the denominators. Step 4: Write the simplified expression including the non-permissible values.
b) Step 1: Factor the numerators and denominators of each expression. Step 3: Multiply the numerators and the denominators together. Cancel common factors. Step 2: Find the non-permissible values. Look at all of the denominators. Step 4: Write the simplified expression including the non-permissible values
c) 18 x 3 5x 15x 2 1 9x 24x 2 2 Step 1: The first step in rational expression problems is always to factor. Step 3: Multiply the numerators and the denominators together. Cancel common factors. Step 2: Find the non-permissible values. Look at all of the denominators. Step 4: Write the simplified expression including the non-permissible values.
DIVIDING RATIONAL EXPRESSIONS 1. Factor the numerators and denominators of each expression. 2. Find the non-permissible values. Look at BOTH of the denominators AND the numerator of the second expression. 3. Take the RECIPROCAL of the SECOND EXPRESSION, and then multiply the numerators and the denominators together. Cancel common factors. 4. Write the simplified expression including the nonpermissible values.
Simplify the following quotient: a) Step 1: Factor. Step 3: Take the reciprocal of the second expression, and then multiply. Cancel common factors. Step 2: Find the non-permissible values. In division problems, use both denominators and the second numerator. Step 4: Write the simplified expression including the non-permissible values
Simplify the following quotient: b) Step 1: Factor. Step 3: Reciprocal, and then multiply. Cancel common factors. Step 2: Find the non-permissible values. Why didn t I use 6w, w + 6, or 9w 2? Step 4: Write the simplified expression including the non-permissible values.
Simplify the following quotient: c) Step 1: Factor. Step 3: Reciprocal, and then multiply. Cancel common factors. Step 2: Find the non-permissible values. Step 4: Write the simplified expression including the non-permissible values.
Simplify the following quotient: c) 2 4x 25 16x 40x 2 3 2x 5 4x 12x Step 1: Factor. Step 3: Reciprocal, and then multiply. Cancel common factors.. Step 2: Find the non-permissible values. Step 4: Write the simplified expression including the non-permissible values.
PG. 238-239, #1, 3, 5, 6, 7, 9. Independent Practice
4.4 ADDING AND SUBTRACTING RATIONAL Chapter 4 EXPRESSIONS
STEPS FOR ADDING OR SUBTRACTING RATIONAL EXPRESSIONS 1. Find a common denominator. Sometimes it will be necessary to factor the denominator first before a common denominator can be found. 2. Find the non-permissible roots. 3. Add or subtract the NUMERATORS as specified in the question. 4. Simplify the resulting expression if possible. This may include factoring the numerator. 5. Write the simplified expression including restrictions.
REMEMBER! ADDING/SUBTRACTING FRACTIONS 2 3 1 2 a) b) c) 5 12 4 6 12 15 7 25
DETERMINING COMMON DENOMINATORS WITH RATIONAL EXPRESSIONS Rational Numbers Common Denominator Rational Expressions Common Denominator 2 3 2 x 1 7 7 x 1 x 1 1 5 3 1 12 6 x 5 4x 20 2 7 3 4 3 2 2 x x 1 5 1 7 1 2 14 6 x 9 4x 12
ADDING/SUBTRACTING RATIONAL EXPRESSIONS WITH THE SAME DENOMINATOR 1. m 2 3 2. m m 15x 4x 2 2 9x 4x
3. 5x 1 3x 1 4. x 1 x 1 3y 7 y( y 2) 4 8y ( y 2) y
5. 1 5x 3 2 2 x 2x x 2x
ADDING/SUBTRACTING RATIONAL EXPRESSIONS WITH DIFFERENT DENOMINATORS Simplify the following sum: a) Find a common denominator. 8x 2 What will we need to multiply 4x by to get 8x 2? 2x Add numerators. 3 2x 2 8x Simplify if possible. Final answer with restrictions. Find the non-permissible values.
b) 9 2 c) 2 2 10y 3 3 2 5y 2b ab
d) 2 t 1 e) 2 5t t 3 2 x 4 x 1
Simplify the following difference: f) Find a common denominator. Common denominator is (2n+1)(n - 3). Subtract numerators. Simplify if possible. Find the non-permissible values: Final answer with restrictions.
FIND A COMMON DENOMINATOR. SOMETIMES IT WILL BE NECESSARY TO FACTOR THE DENOMINATOR(S) FIRST Simplify the following expression: g) 9 2 x 9 3 2x 6
Simplify the following expression: h) 2 x x 7 14 3x 3x 21
SIMPLIFY THE FOLLOWING EXPRESSION: I) 5x 1 2x 6 x 16x 2 9
SIMPLIFY THE FOLLOWING EXPRESSION: J) 2x 3 3x 3 5x 3 3x 2 3x
PG. 249-250, #4, 5, 6, 7, 8, 13 Independent practice
EXAMPLE A jet flew along a straight path from Calgary to Vancouver, and back again, on Monday. It made the same trip on Friday. On Monday, there was no wind. On Friday, there was a constant wind blowing from Vancouver to Calgary at 80 km/h. While travelling in still air, the jet travels at a constant speed. Determine which trip took less time. What is the equation for time, when you have speed and distance? How fast is the jet s airspeed for two different trips on Friday?
EXAMPLE CONTINUED Recall that when you add fractions you need to have a common denominator. Total for Friday Total for Monday
EXAMPLE CONTINUED To be able to compare these two times, we need to have either the numerator or the denominator be the same. Can we easily create a common numerator or denominator? Which denominator is larger? So what does that say about T 1? A larger denominator means that the fraction is smaller. So that means that T 1 is a smaller number, and was a shorter trip.
P. 249-250, #9, 10, 11. Independent practice
4.5 SOLVING RATIONAL EQUATIONS Chapter 4
IN THIS SECTION, YOU WILL Solve equations containing rational expressions. **It is intended that all rational expressions be simplified to linear and quadratic equations. Check if the solutions are permissible. **Roots that are non-permissible values are called extraneous roots.
STEPS FOR SOLVING RATIONAL EQUATIONS 1. Factor if you can. 2. Find the non-permissible values. 3. Multiply every term in the equation by the LCD (lowest common denominator).this will enable you to eliminate fractions from the equation. 4. Solve the resulting linear or quadratic equation. 5. Check your solution. If a root of a rational equation is a non-permissible value of the rational expressions in the original equation, it is an extraneous root and must be dismissed as a valid solution.
SOLVE THE FOLLOWING EQUATIONS: A. Non-permissible values: LCD 3 5 x 4 4x 13 x x 0 Multiply through by LCD And cancel where possible 3 5 13 4x 4 x 4x x 4 x Solve for x: 4( 3) x( 5) 4( 13) 12 5x 52 5x 52 12 5x 40 8 Check your solution: 3 8 5 4 x 13 8
B. 3 7 x 2x 1 5
2 C. 2x 1 x 2 x 3 4 x 3
D. 5 x x 1 4 5 x 1
E. x x 3 2 3 x 3
F. 1 3x 1 2 x 2 x 4 x 2
G. 2 2 a 2 a 2 a 4 4 a a 2
H. 14 7 6 2 x 2x x 2 x
WHEN WE HAVE ONE RATIONAL EXPRESSION ON EITHER SIDE OF AN EQUAL SIGN, WE MAY USE CROSS-MULTIPLICATION IF WE PREFER. I. 5 3 x 3 x 1
J. 3 1 2 x 4x x 4
P. 258, #1, 5AB, 6. Independent practice
JUST TO RECAP! Solve the following equation for x: You can tell that it is an equation problem and not an expression problem because of the equal sign. There is a different set of rules, so it s important to differentiate. Step 1: Factor. Step 4: Multiply each numerator by the whole LCD. Step 5: Simplify and Solve. You should be able to get rid of the denominators. Step 2: Non-permissibles. Step 3: What would the LCD be? However, 3 is a NPV. What does this mean?
APPLICATIONS OF RATIONAL EXPRESSIONS In this section, you should be able to: Write an equation to represent a problem. Solve the equation by multiplying through by the LCD.
EXAMPLE The sum of a number and its reciprocal is. 5 2
A COMMON WORD PROBLEM THAT IS OFTEN ASKED INVOLVES SHARED TASKS: Jack can paint a room in a hours, Jill can paint a room in b hours. Working together they can paint a room in t hours. The format of these equations is always the same: 1 1 1 a b t Remember! Sometimes we will get inadmissible roots when problem solving. For example, it does not make sense to have negative time, negative distance, etc
EXAMPLES OF SHARED TASKS 1. Sheri can mow a lawn in 4 hours. Mary can mow the same lawn in 5 hours. How long would it take both of them working together to the mow that lawn? 1 4 1 5 1 t Solve for t: LCD = 20t
2. TERRY CAN WAX THE FLOOR OF THE SCHOOL GYM IN 8 HOURS. MIMI TAKES 6 HOURS TO WAX THE GYM FLOOR. HOW LONG WILL IT TAKE TO WAX THE GYM FLOOR IF THEY WORK TOGETHER?
3. IT TAKES MIKE 9 HOURS LONGER THAN JASON TO CONSTRUCT A FENCE. WORKING TOGETHER, THEY CAN CONSTRUCT THE FENCE IN 20 HOURS. HOW LONG WOULD IT TAKE MIKE IF HE WAS WORKING ALONE?
4. GERARD TAKES 5 HOURS LONGER THAN HUBERT TO ASSEMBLE A PLAY SET. WORKING TOGETHER, THEY COULD ASSEMBLE THE SET IN 6 HOURS. DETERMINE HOW LONG IT WOULD TAKE EACH PERSON IF THEY WORKED ALONE.
5. When they work together, Stuart and Lucy can deliver flyers to all the homes in their neighbourhood in 42 minutes. When Lucy works alone, she can deliver the flyers in 13 minutes less time than Stuart when he works alone. When Stuart works alone, how long does he take to deliver the flyers? Let y be the time it takes Stuart alone. then how long does it take for Lucy? Lucy takes (y 13) Always consider the fraction of deliveries that can be made in 1 minute: Stuart alone: Lucy alone: Together: Does y = 6 make sense?
OTHER TYPES OF WORD PROBLEMS Rima bought a case of concert T-shirts for $450. She kept two T-shirts for herself and sold the rest for $560, making a profit of $10 on each T-shirt. How many T-shirts were in the case? What is the expression for price per t-shirt? What is the expression for profit per shirt? Do both these answers make sense?
7. AMBER DRIVES 720 KM FROM CORNER BROOK TO ST. JOHN S. ON HER WAY BACK TO CORNER BROOK, SHE REDUCES HER SPEED BY 10 KM/H BECAUSE OF POOR WEATHER CONDITIONS. THE RETURN TRIP TAKES 1 HOUR LONGER. WHAT WAS HER AVERAGE SPEED ON HER WAY TO ST. JOHN S? Let x represent her average speed. Solve for x: Time from CB to St. J: 720 x Time from St. J to CB: 720 x 10 Dif ference in times: 720 x 10 720 x 1
PG. 259-260, #8, 10, 11, 12, 14, 15 Independent practice
CHALLENGE YOURSELF! Solve the equation. What are some non-permissible values? 4k -1 k + 2 - k + 1 k - 2 = k2-4k + 24 k 2-4 4k -1 k + 2 - k + 1 k - 2 = k2-4k + 24 k 2-4 The non-permissible values are 2 and -2. Þ 4k - 1 k + 2 - k + 1 k - 2 = k2-4k + 24 (k - 2)(k + 2) æ Þ (k - 2)(k + 2) è ç 4k - 1 k + 2 - k + 1 ö æ k 2-4k + 24 ö k - 2ø = (k - 2)(k + 2) è ç (k - 2)(k + 2) ø æ Þ (k - 2)(k + 2) è ç 4k - 1ö æ k + 2 ø - (k - 2)(k + 2) è ç k + 1ö æ k 2-4k + 24 ö k - 2ø = (k - 2)(k + 2) è ç (k - 2)(k + 2) ø
EXAMPLE Solve the equation. What are some non-permissible values? 4k -1 k + 2 - k + 1 k - 2 = k2-4k + 24 k 2-4 Þ (k - 2) ( 4k -1) - (k + 2) ( k +1) = k 2-4k + 24 Þ (4k 2-9k + 2) - (k 2 + 3k + 2) = k 2-4k + 24 Does k = 2 work? Why or why not? Þ 3k 2-12k = k 2-4k + 24 Þ 2k 2-8k - 24 = 0 Þ (k - 6)(k + 2) = 0 Þ k = 6 Þ k = -2 Þ k 2-4k -12 = 0