Unit 4 - Rational expressions and equations. Rational expressions. A Rational Number is the ratio

Transcription

1 Unit 4 - Rational expressions and equations 4.1 Equivalent rational expressions Rational expressions What is a rational number? A Rational Number is the ratio Examples: What might a rational expression be? A rational expression is an 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 where. Examples: Are the following rational expressions? Why or why not? 4 5 x x 4 y x 1 5 x x y 4 1 P a g e

2 Non-permissible values What value can x not have? 1 x For all rational expressions with variables in the, we need to define the non-permissible values. These are the values for a variable that makes an expression o In a rational expression, this is a value that results in a These non-permissible values are the on the domain of the rational expression. Determine the non-permissible values for following: A) 5 4x x x B) C) 4 x 4 x x x 7 E) x 9 D) x 4 x ( x 1)( x ) F) 1 1 x x P a g e

3 Practice: What are the non-permissible roots? 5x 5x A) B) 5x 16 x 8 5x C) 0x 15x D) 5x x 5 3 P a g e

4 Write a rational expression that has non-permissible values of: A) and 4 B) 0, -, and 3 1 C) D) 4 3 and 4 7 Non-Permissible vs. Inadmissible Values Non-Permissible Values: values of a variable that Inadmissible Values: values of a variable that Example: If a boat travelled 0 km with a speed of x km/h, the time taken for the trip would be represented by the expression x 0. x 0 is a since it makes the denominator = 0. Negative x-values are since they result in negative time (which doesn t make sense!) Independent Practice Pages 3-4 #3(find non-permissible values only), 9a (i, ii,),10, 11a (i, ii,), 16 4 P a g e

5 Equivalent Fractions Writing Equivalent Fractions involving only numbers: Rational Numbers To find equivalent rational numbers (fractions), we Example: Write two rational numbers that are equivalent to: 8 A ) 1 5 B ) 100 Equivalent Rational Expressions (Fractions with variables) To find equivalent rational expressions, we both the numerator and denominator by the same number. x 3 o EX. x o Do not multiply by a factor that introduces a restriction. o That is, do not multiply by something like: o EX. What happens to the restrictions when x 3 x is multiplied by 5 P a g e

6 Equivalent Rational Expressions must have the. However we can divide the numerator and denominator by a provided we keep the same x x 1 Ex. Consider: x x 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 two equivalent rational expressions. A) 4 x 6 P a g e

7 B) Write a rational expression that is equivalent to 4x 8 x 4x Firstly, what are the non-permissible values? Can I possibly factor either the numerator or the denominator? We need to write our non-permissible values at the end. Write a second expression that is equivalent to 4x 8 x. What 4x happens when we multiply by x? Is the new equation an equivalent x expression? C) 6n 3 3 n 1 7 P a g e

8 . For each of the following, determine if the rational expressions are equivalent. A) 9 3x 1 and 18 6x Two conditions in order for the expressions to be equivalent: If you multiply or divide by a common factor, you should get B) x 4x 1 and x x Another method is to check using. So, choose a value that you d like to put in for x. Let s try 8 P a g e

9 3. Summary: To determine whether two rational expressions are equivalent: The numerator and denominator of one expression must be multiplied or divided by the to produce the other expression The two expressions must have the (same non-permissible values). Substitution: We can use substitution to show that two rational expressions are equivalent. If we can come up with 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 be used to show that two rational expressions ARE equal. Expressions may be equal for some values but not necessarily all values. 9 P a g e

10 Examples: A) Explain why 3 and x 3( x 1) x( x 1) are NOT equivalent expressions. B) Using Substitution, explain why 5x 4 x and 3x x are NOT equivalent expressions. (Use the values x 1 and x ) Independent Practice Pg. 3-4, #1, 3, 5, 6, 14, 15, 10 P a g e

11 4. Simplifying Rational Expressions Simplifying rational expressions is similar to simplifying algebraic fractions. Example: Simplify each fraction A. B. C Simplifying rational expressions results in an expression that is easier to work with than the original: Simplify the following rational expression: What are both 4 and 18 divisible by? Simplify the following rational expression: Can I factor the numerator? 11 P a g e

12 Steps for Simplifying Rational Expressions 1. Find the non-permissible values.. 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. Simplify the following rational expression: A. B. 6x 3 8x 4 1 P a g e

13 C. 6x 7x 4x 18x 3 D. x 9 x 6x E. 3x 1 6x 1 13 P a g e

14 F. 4 5x x 30 G. 3 16x 1x 3 3x 18x H. 4x 6x 4 16x P a g e

15 Identify and correct any errors in the following simplifications: A. = = B. = C. Independent Practice Pg. 9-31, #3, 4, 5, 8, 9, 10, P a g e

16 4.3 Multiplying and Dividing Rational Expressions Steps for multiplying rational expressions 1. Factor the numerators and denominators of each expression.. 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 non-permissible 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 : Find the non-permissible values. Look at all of the denominators. Step 4: Write the simplified expression including the non-permissible values. 16 P a g e

17 b) Step 1: Factor the numerators and denominators of each expression. Step 3: Multiply the numerators and the denominators together. Cancel common factors. Step : Find the non-permissible values. Look at all of the denominators. Step 4: Write the simplified expression including the non-permissible values. c) 3 18x 5x 15x 1 9x 4x Step 1: Factor the numerators and denominators of each expression. Step 3: Multiply the numerators and the denominators together. Cancel common factors. Step : Find the non-permissible values. Look at all of the denominators. Step 4: Write the simplified expression including the non-permissible values. 17 P a g e

18 Dividing Rational Expressions 1. the numerators and denominators of each expression.. Find the values. Look at of the AND the of the second expression. 3. Take the of the, and then the numerators and the denominators together. common factors. 4. Write the simplified expression including the non-permissible values. Simplify the following quotients: a) Step 1: Factor the numerators and denominators of each expression. Step 3: Take the reciprocal of the second expression, and then multiply. Cancel common factors. Step : 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. 18 P a g e

19 b) Step 1: Factor the numerators and denominators of each expression. Step 3: Take the reciprocal of the second expression, and then multiply. Cancel common factors. Step : 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. c) Step 1: Factor the numerators and denominators of each expression. Step 3: Take the reciprocal of the second expression, and then multiply. Cancel common factors. Step : 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. 19 P a g e

20 d) 4x 5 16x 40x Step 1: Factor. 3 x 5 4x 1x Step 3: Reciprocal, Multiply, Cancel. Step : Find the non-permissible values. Step 4: Write the simplified expression including the non-permissible values. Independent Practice Pg , #1, 3, 5, 6, 7, 9. 0 P a g e

21 4.4 Adding and Subtracting Rational Expressions Steps for adding or subtracting rational expressions 1. Find a. it will be necessary to the denominator before a common denominator can be found.. Find the. 3. as specified in the question. 4. the resulting expression if possible. This may include factoring the numerator. 5. Write the simplified expression. Remember! Adding/subtracting fractions a) 1 b) 5 4 c) P a g e

22 Determining Common Denominators of Rational Expressions Rational Numbers Common Denominator Rational Expressions Common Denominator x 1 x 1 x x 5 4x x x x 9 4x 1 Adding/Subtracting Rational Expressions with the Same Denominator 1. m 3 m m. 15x 9x 4x 4x P a g e

23 3. 5x 1 3x 1 x 1 x y y( y 7 4 8y ) ( y ) y x 3 x x x x Adding/Subtracting Rational Expressions with Different Denominators Simplify the following sums and differences: a) What will we need to multiply 4x by to get 8x? 3 P a g e

24 9 b) 10y 5y c) 3 3 b ab 5t t 1 t d) e) 3 4 x x 1 4 P a g e

25 f) Find a common denominator. Subtract numerators. Simplify if possible. The non-permissible values: Final answer with restrictions. Find a common denominator. Sometimes it will be necessary to factor the denominator(s) first. 5 P a g e

26 g) 9 3 x 9 x 6 h) x 7 3x x 14 3x 1 6 P a g e

27 i) 5x 1 x 6 16x x 9 j) x 3 5x 3 3x 3 3x 3x Independent practice Pg , #4, 5, 6, 7, 8, 13 7 P a g e

28 4.5 Solving Rational Equations In this section, you will ** It is intended that all rational expressions be simplified to. ** Roots that are non-permissible values are called. Steps for Solving Rational Equations 1. Factor if you can.. Find the non-permissible values. 3. in the equation by the.this will enable you to from the equation. 4. the resulting. 5. your solution. If a root of a rational equation is a value of the rational expressions in the original equation, it is an and must be. 8 P a g e

29 Solve the following equations: A. 3 5 x 4 13 x B x x 5 9 P a g e

30 C. x 1 x x 3 4 x 3 D. 5x 5 4 x 1 x 1 30 P a g e

31 E. x x 3 3 x 3 F. 1 3x 1 x x 4 x 31 P a g e

32 G. a 4 a a a 4 a H x x x x 3 P a g e

33 When we have one rational expression on either side of an equal sign, we may use cross-multiplication if we prefer. I. x x 1 J. 3 1 x 4x x 4 Independent Practice P. 58, #1, 5Ab, 6 33 P a g e

34 Just to recap! Solve the following equation for x: You can tell that it is an equation problem and not an expression problem because. There is a different set of rules, so it s important to differentiate. 34 P a g e

35 Applications of Rational Expressions In this section, you should be able to: the equation by. Example The sum of a number and its reciprocal is P a g e

36 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: Remember! Sometimes we will get when problem solving. For example, it to have negative time, negative distance, etc 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? 36 P a g e

37 . 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 0 hours. How long would it take Mike if he was working alone? 37 P a g e

38 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. 38 P a g e

39 5. When they work together, Stuart and Lucy can deliver flyers to all the homes in their neighbourhood in 4 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? 39 P a g e

40 Other types of word problems 6. 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? 40 P a g e

41 7. Amber drives 70 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. Time from CB to St. J: Time from St. J to CB: Difference in times: Solve for x: Independent practice Pg , #8, 10, 11, 1, 14, P a g e

42 Challenge Yourself! Solve the equation. What are some non-permissible values? 4k -1 k + - k + 1 k - = k - 4k + 4 k P a g e