Slide 1 / 179 Slide 2 / 179 Algebra II Rational Expressions & Equations 2015-08-15 www.njctl.org Slide 3 / 179 Slide 4 / 179 Table of Contents click on a topic to go to that section Working with Rational Expressions Joint and Inverse Variation Simplifying Rational Expressions Working with Rational Multiplying Rational Expressions Expressions Dividing Rational Expressions Adding and Subtracting Rational Expressions Solving Rational Equations Applications of Rational Equations Graphing Rational Functions Return to Table of Contents Slide 5 / 179 Slide 6 / 179 What is a rational expression? Goals and Objectives Students will simplify rational expressions, as · well as be able to add, subtract, multiply, and A rational expression is the ratio of two polynomials. It is divide rational expressions. written as a fraction with polynomial expressions in the Students will solve rational equations and use · numerator and denominator. them in applications. Students will graph rational functions and · identify their holes, vertical asymptotes, and horizontal asymptotes.
Slide 7 / 179 Slide 8 / 179 Why do we need this? Rational expressions are often used to simplify expressions with long polynomials in both the numerator and denominator. Since it is more efficient to work with simple problems and situations, knowing how to simplify rational expressions makes looking at graphs and other problems easier. Rational expressions and equations are often used to model more complex equations in fields such as science and engineering. Rational expressions are applicable in working with forces and fields in physics and aerodynamics. Slide 9 / 179 Slide 10 / 179 Inverse and Joint Variation Return to Table of Contents Slide 11 / 179 Slide 12 / 179 Variation Inverse Variation Variation describes the relationship between variables. With Inverse variation, when one element increases, the other element decreases. Or, vice versa, when one element decreases, the other element increases. There are three types of variation: Examples: direct, inverse and As you pull on a As you increase As you increase joint variation. rubber band to your altitude by your amount of make it longer, hiking up a spending, you Each type describes a different relationship. the width of the mountain, you decrease the band gets will feel a amount of money smaller. decrease in the available to you. temperature.
Slide 13 / 179 Slide 14 / 179 Variation Joint Variation Using more mathematical vocabulary... Joint variation is the same as direct variation, but is used when two or more elements affect what another element does. If one or both elements increase, the other element increases. Or, vice versa, when Inverse variation: The temperature of the air varies one or both elements decrease, the other element also decreases. inversely with the altitude. written as: Examples: As you increase As you either As you increase the radius and/ decrease the speed the length or or the height of you drive or width of your a cone, you decrease the time Joint variation: The volume of a cone varies jointly backyard fence, increase the you drive, you will with the square of its radius and its height. you increase the volume. decrease the area of your distance you cover. written as: backyard. Slide 15 / 179 Slide 16 / 179 Variation Variation Steps to solving a variation problem: Notice that in each of these variations there is an additional number whose value does not change: 1) Determine an equation based on each type of variation. Inverse: y = k / x Joint: y = kxz 2) Find the constant of variation (k) 3) Rewrite the equation substituting a value for k. 4) Use the final equation to find the missing value. This number is called the constant of variation and is denoted by k . Slide 17 / 179 Slide 18 / 179 Variation Variation Example: Example: The volume of a square pyramid varies jointly with the area of the base (s 2 ) and the height. If the volume is 75 when If y varies inversely with x, and y = 10 when x = 4, find x when y = 80. the base side is 5 and the height is 9, find the volume when the height is 12 and the base side is 4.
Slide 19 / 179 Slide 20 / 179 1 If y varies inversely with x, and y = 10 2 If y varies inversely with x, and y = 3 when x = 15, when x = -4, find y when x = 8. find y when x = 5. Slide 21 / 179 Slide 22 / 179 4 If y varies jointly with x and z, and y = 3 when x = 4 and 3 If y varies jointly with x and z, and y = 6 when z = 6, find y when x = 6 and z = 8. x = 3 and z = 9, find y when x = 5 and z = 4. Slide 23 / 179 Slide 24 / 179 Simplifying Rationals A rational expression is an expression that can be written in the form , where a variable is in the denominator. Simplifying Rational Expressions The domain of a rational expression is all real numbers excluding those that would make the denominator 0. (This is very important when solving rational equations.) For example, in the expression , 2 and -2 are Return to restricted from the domain. Table of Contents
Slide 25 / 179 Slide 26 / 179 Slide 27 / 179 Slide 28 / 179 Simplifying Rationals 5 Simplify Remember to use properties of exponents and/or factoring to simplify the rational expressions. B C A D Slide 29 / 179 Slide 30 / 179 6 Simplify 7 Simplify A B C D A C D B
Slide 31 / 179 Slide 32 / 179 Slide 33 / 179 Slide 34 / 179 Multiplying Rational Expressions Return to Table of Contents Slide 35 / 179 Slide 36 / 179
Slide 37 / 179 Slide 38 / 179 Multiply Slide 39 / 179 Slide 40 / 179 10 Simplify A C D B Slide 41 / 179 Slide 42 / 179
Slide 43 / 179 Slide 44 / 179 Dividing Rational Expressions Return to Table of Contents Slide 45 / 179 Slide 46 / 179 14 Simplify 15 Simplify A C A C B D D B Slide 47 / 179 Slide 48 / 179
Slide 49 / 179 Slide 50 / 179 18 Simplify Adding and Subtracting Rational Expressions A C D B Return to Table of Contents Slide 51 / 179 Slide 52 / 179 Adding and Subtracting Rational Expressions Just as in multiplication and division, when adding or subtracting rationals, use the same rules as basic fractions. Recall: When adding and subtracting fractions, you MUST use common denominators. Slide 53 / 179 Slide 54 / 179
Slide 55 / 179 Slide 56 / 179 Adding and Subtracting Rational Expressions To add and subtract rational expressions they must have common denominators. Identify the LCD and rewrite the rational expressions with the same denominator. Example: No common denominator. x 2 is the least common denominator.(LCD) Multiply by an expression equal to 1. (Multiply numerator and denominator by the same quantity.) Then Add. Simplify if possible. Slide 57 / 179 Slide 58 / 179 Adding and Subtracting Rational Expressions Example Solve: Step 1: Find LCD Step 1: LCD = ( x + 2)( x - 2) Step 2: Multiply each term by an expression equal Step 2: to 1 to obtain LCD for each term. Step 3: Step 3: Add or subtract numerators Step 4: Simplify Step 4: Slide 59 / 179 Slide 60 / 179 Example Solve: The denominators are additive inverses. Step 1: Step 2: Step 3: Step 4:
Slide 61 / 179 Slide 62 / 179 Example Continued Click Remember: you can always check results by substituting values for the variables, being sure to avoid values for which the expression is undefined. Slide 63 / 179 Slide 64 / 179 Common Denominator Find the LCDs for the following. Describe any restrictions on the variables. Slide 65 / 179 Slide 66 / 179 23 Simplify A C B D
Slide 67 / 179 Slide 68 / 179 25 Simplify 24 Simplify A C C A B D D B Slide 69 / 179 Slide 70 / 179 Division by Zero Find an equivalent rational expression in lowest terms, and identify the value(s) of the variables that must be excluded to prevent division by zero. ( Derived from ( Slide 71 / 179 Slide 72 / 179 Equivalent Expressions Determine whether or not the rational expressions below are equivalent for . Explain how you know. ( Derived from (
Slide 73 / 179 Slide 74 / 179 Adding and Subtracting Rationals Problem is from: Click for link for commentary and solution. Solving Rational Equations Return to Table of Contents Slide 75 / 179 Slide 76 / 179 Solving Rational Equations Step 1: Find LCD Step 2: Multiply EACH TERM by LCD Step 3: Simplify Step 4: Solve Step 5: Check for Extraneous Solutions Slide 77 / 179 Slide 78 / 179 Example Continued Step 5: Explanation When the solution of h = 2 is substituted into the original equation, it creates two undefined terms: This means that h = 2 is an extraneous solution and the rational equation has no solution.
Slide 79 / 179 Slide 80 / 179 Slide 81 / 179 Slide 82 / 179 Solving Rational Equations 27 Use Steps 1 - 4 to solve for x : Example: Remember to find LCD and check all solutions. C 24 A -9 D 30 B 9 Slide 83 / 179 Slide 84 / 179 28 Is the solution to the previous question valid when 29 Use Steps 1 - 4 to solve for m : substituted into the original equation? A Yes, the solution is valid. Answer C 5 A -12 B No, the solution creates a false mathematical statement and is therefore an extraneous solution. C No, the solution creates an undefined term(s) and is B -5 D 12 therefore an extraneous solution.
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