13.1 Understanding Piecewise-Defined Functions

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1 Name Class Date 13.1 Understanding Piecewise-Defined Functions Essential Question: How are piecewise-defined functions different from other functions? Resource Locker Eplore Eploring Piecewise-Defined Function Models A piecewise function has different rules for different parts of its domain. The following situation can be modeled b a piecewise function. Armando drives from his home to the grocer store at a speed of 0.9 mile per minute for minutes, stops for minutes to bu snacks, and then drives to the soccer field at a speed of 0.7 mile per minute for 3 minutes. The graph shows Armando s distance from home. Houghton Mifflin Harcourt Publishing Compan Image Credits: Steve Cukrov/Shutterstock The three different sections in the graph show that there are three different parts to the function. The function for this graph will have three function rules, each for a different range of values for. The following steps can be used to build the piecewise function to model this situation. B Distance (mi) Determine the function rule for the first minutes for the domain, 0. The rate of change is Armando s speed in miles per minute. mi m = _ = min 0 The -intercept is. 6 Time (minutes) The function for the first minutes is =. Determine the function rule for the net minutes for the domain, < 6. mi Because the distance is not changing, the rate of change is m = _ =. min Module Lesson 1

2 The function for the net minutes is a constant function. The constant -value is Armando s distance from home at the end of the first minutes. = 0.9 = 0.9 () = C Determine the function rule for the last 3 minutes. The rate of change is Armando s speed in miles per minute. mi m = _ = min The point at which the function begins is (, ). Use the point and slope to construct the function rule for the last 3 minutes. - = ( - ) - = - = - D Use all three parts to build the piecewise function that represents this situation. if 0 ƒ () = if < 6 if 6 < 9 Reflect 1. Discussion Describe how the domain is constructed so that the piecewise function is a function with no more than one dependent variable for an independent variable. Eplain 1 Evaluating Piecewise-Defined Functions The greatest integer function is a piecewise function whose rule is denoted b, which represents the greatest integer less than or equal to. The greatest integer function is an eample of a step function, a piecewise function in which each function rule is a constant function. To evaluate a piecewise function for a given value of, substitute the value of into the rule for the part of the domain that includes. Houghton Mifflin Harcourt Publishing Compan Module Lesson 1

3 Eample 1 Evaluate each piecewise function for the given values. Find ƒ (-3), ƒ (-.9), ƒ (0.7), and ƒ (1.06) for ƒ () =. The greatest integer function ƒ () = can also be written in the form below. ƒ () = -3 if -3 < - - if - < -1-1 if -1 < 0 0 if 0 < 1 1 if 1 < if < 3-3 is in the interval -3 < -, so ƒ (-3) = is in the interval -3 < -, so ƒ (-.9) = is in the interval 0 < 1, so ƒ (0.7) = is in the interval 1 <, so ƒ (1.06) = 1. B Find ƒ (-3), ƒ (-0.), ƒ (0), and ƒ () for ƒ () = - if < if 0-3 < 0, so ƒ (-3) = - (-3) = 0 0, so ƒ (0) = + 1 = -0. < 0, so ƒ (-0.) = = 0, so ƒ () = + = Reflect. For positive numbers, how is appling the greatest integer function different from the method of rounding to the nearest whole number? Houghton Mifflin Harcourt Publishing Compan Your Turn 3. Find ƒ (-), ƒ (-0.), ƒ (3.7), and ƒ (5) for ƒ () = Eplain - if < + 3 if <. if Graphing Piecewise-Defined Functions You can graph piecewise-defined functions to illustrate their behavior. Eample Graph each function. - if < 0 ƒ () = + 1 if 0 Make a table of values f () Module Lesson 1

4 B ƒ () = Make a table of values f () Reflect. Wh does the graph in Eample A use ras and not lines? 5. Use the graph of the greatest integer function from Eample B to eplain wh this function is called a step function. Your Turn if < 6. ƒ () = + 3 if < if Houghton Mifflin Harcourt Publishing Compan Module Lesson 1

5 Eplain 3 Modeling with Piecewise-Defined Functions Some real-world situations can be described b piecewise functions. Eample 3 Write a piecewise function for each situation. Then graph the function. Travel On her wa to a concert, Maisee walks at a speed of 0.03 mile per minute from her car for 5 minutes, waits in line for a ticket for 3 minutes, and then walks to her seat for minutes at a speed of 0.01 mile per minute. Epress the Maisee s distance traveled d (in miles) as a function of time t (in minutes). For 0 t 5, m = 0.03 and b = 0, so d (t) = 0.03t. For 5 < t, m = 0 and b = 0.15, so d (t) = For < t 1, m = 0.01 beginning at (, 0.15), so d (t) = 0.01t d (t) = 0.03t if 0 t if 5 < t 0.01t if < t 1 Distance (miles) d(t) t 6 10 Time (minutes) B Travel On his wa to class from his dorm room, a college student walks at a speed of 0.05 mile per minute for 3 minutes, stops and talks to a friend for 1 minute, and then to avoid being late for class, runs at a speed of 0.10 mile per minute for minutes. Epress the student s distance traveled d (in miles) as a function of time t (in minutes). For 0 t 3, m = and b = 0, so d (t) = t. For 3 < t, m = 0 and b = 0.15, so d (t) = For < t 6, m = beginning at, so = -. Houghton Mifflin Harcourt Publishing Compan Your Turn d (t) = t if 0 t if 3 < t t - if < t 6 d(t) t Time (minutes) 7. Finance A savings account earns 1.% simple interest annuall for balances of $100 or less,.% simple interest for balances greater than $100 and up to $500, and 3.% simple interest for balances greater than $500. Write a function rule for the interest paid b the account and graph the function. Distance (miles) Annual Interest ($) P(d) 16 1 d Balance ($100s) Module Lesson 1

6 Eplain Building Piecewise-Defined Functions from Graphs You can find the function rules for a piecewise function when ou are given the graph of the function. Eample Write an equation for each graph Find the equation of the ra on the right. m = _ = _-3-1 = 3 Because the point (1, 1) is on the ra, - 1 = 3( - 1), so = 3 - The equation of the line that contains the horizontal ra is = -1. The equation for the function is = -1 if < if 1 B Find the equation for the ra on the left m = _ = _ = -1 - Because the point (-1, 1) is on the ra, = ( ), so = The equation of the horizontal ra is =. The equation for the function is = + if -1 if > -1 Your Turn Elaborate 9. How are the greatest integer function and ƒ () = related? Houghton Mifflin Harcourt Publishing Compan 10. Essential Question Check-In How man function rules do functions that are not piecewise-defined have? Module 13 5 Lesson 1

7 Evaluate: Homework and Practice Evaluate each piecewise function for the given values. 1. Find ƒ (-), ƒ (-3.1), ƒ (1.), and ƒ (.) for ƒ () =. Online Homework Hints and Help Etra Practice. Find ƒ (-3), ƒ (-.1), ƒ (0.6), and ƒ (3.3) for ƒ () = 3. Find ƒ (-), ƒ (-.9), and ƒ (1.9) for ƒ () = if 0 if 0 < < 1. if 1-5 if -3 + if -3 < if 0. Find ƒ (-6), ƒ (-.), ƒ (1.) and ƒ (3.6) for ƒ () = Find ƒ (-3), ƒ (-1), and ƒ (1) for ƒ () = _ if - if - < 0 1 if 0-6. Find ƒ (-), ƒ (-1), ƒ (0), ƒ (), and ƒ (9) for ƒ () = + 6 _ + if - if - < < if if >. 7. Find ƒ (-.), ƒ (-1.), ƒ (0.), and ƒ (1.6) for ƒ () = Houghton Mifflin Harcourt Publishing Compan. if 0 Find ƒ (0), ƒ (), and ƒ () for ƒ () = 0 if > 0. Graph each piecewise function if < 0 ƒ () = if if < ƒ () = - if Module Lesson 1

8 11. ƒ () = ƒ () = Write an equation for each graph Houghton Mifflin Harcourt Publishing Compan Module 13 5 Lesson 1

9 Write a piecewise function for each situation. Then complete the table and the graph. 17. Finance A garage charges the following rates for parking (with an hour limit): $ per hour for the first hours $ per hour for the net hours No additional charge for the net hours Epress the cost C (in dollars) as a function of the time t (in hours) that a car is parked in the garage. C (t) = t C (t) t C (t) 1. Cost Analsis The cost to send a package between two cities is $.00 for an weight less than 1 pound. The cost increases b $.00 when the weight reaches 1 pound and again each time the weight reaches a whole number of pounds after that. Cost (dollars) 1 0 C(t) 6 Time (hours) t Houghton Mifflin Harcourt Publishing Compan Kues/Shutterstock Epress the shipping cost C (in dollars) as a function of the weight (in pounds). Epress our answer in terms of the greatest integer function w. C (w) = w C (w) Cost (dollars) 16 0 C(w) 6 Weight (pounds) w Module Lesson 1

10 19. Golfing A local golf course charges members $30 an hour for the first three hours, $35 an hour for the net five hours, and nothing for the last hours, for a maimum of 10 hours. Epress the cost C (in dollars) as a function of the time t C(t) (in hours) that a member plas golf at this golf course. 0 C (t) = Cost (dollars) t 6 10 C (t) Time (hours) t 0. Construction A construction compan is building a new parking garage and is charging the following rates: $5000 a month for the first months; $000 a month for the net months; $6000 in total for the last months, when the garage will be completed. This amount will be paid in a lump sum at the end of the 6th month. Epress the cost C (in thousands of dollars) as a function of the time t (in months) that the construction compan works on the parking garage. C (t) = t 6 10 C (t) Cost (thousands of dollars) C(t) 6 Time (months) t 1. State the domain and range of each piecewise function. A. = 5 if 0 < 1 0 if > 1 D. = + 1 if - 3 if - < < if - if 0 B. = if > 0 C. = if 3 if > 1 if -3 E. = 1 if -3 < < 1 if Houghton Mifflin Harcourt Publishing Compan Module Lesson 1

11 H.O.T. Focus on Higher Order Thinking. Critical Thinking Rewrite the piecewise function into a function of the greatest integer function. -6 if - < -1-3 if -1 < 0 f () = 0 if 0 < 1 3 if 1 < 6 if < 3 3. Eplain the Error Clara was given the following situation and told to write a piecewise function to describe it. While eercising, a person loses weight in the following manner: 0.5 pound per hour for the first hour 0.7 pound per hour for the net three hours 0.1 pound per hour until the workout is finished Clara produced the following result. What did she do wrong and what is the correct answer? 0.5t if 0 t 1 W (t) = 0.7t if 1 t < 0.1t if t Houghton Mifflin Harcourt Publishing Compan. Critical Thinking Write an equation for the shown graph. Epress the answer in terms of. 5. Communicate Mathematical Ideas Is a piecewise function still a function if it contains a vertical line? Eplain wh or wh not Module Lesson 1

12 Lesson Performance Task Suppose someone is traveling from New York Cit to Miami, Florida. The following table describes the average speeds at various intervals on this 100-mile trip. Distance Traveled (hundreds of miles) Average Speed (mi/h) 0 < d 37.7 < d 6.6 < d < d 5.5 < d < d A. Graph the distance function. Make sure to use appropriate labels s d B. Write the piecewise function that is given b the table. C. Suppose the destination was changed from Miami, Florida to Minneapolis, Minnesota instead. Eplain wh it is not oka to use the piecewise function created for the trip from New York to Miami when traveling to Minneapolis, even though the distance is comparable. Houghton Mifflin Harcourt Publishing Compan Module 13 5 Lesson 1

13 Name Class Date 13. Absolute Value Functions and Transformations Essential Question: What are the effects of parameter changes on the graph of = a - h + k? Resource Locker Eplore Understanding the Parent Absolute Value Function The most basic absolute value function is a piecewise function given b the following rule. ƒ () = = if 0 - if < 0 This function is sometimes called the parent absolute value function. Complete each step to graph this function. A B Complete the table of values. f () = Plot these points on a graph and using two ras, connect them to displa the absolute value function. Houghton Mifflin Harcourt Publishing Compan C The verte of an absolute value function is the single point that both ras have in common. Identif the verte of the parent absolute value function. Module Lesson

14 Reflect 1. What is the domain of ƒ () =? What is the range?. For what values of is the function ƒ () = increasing? decreasing? Eplain 1 Graphing Translations of Absolute Value Functions You can compare the graphs of absolute value functions in the form g () = - h + k, where h and k are real numbers, with the graph of the parent function ƒ () = to see how h and k affect the parent function. Eample 1 Graph each absolute value function with respect to the parent function ƒ () =. A g () = First, create a table of values for and g(). g () = Now graph the function along with the parent function. - 0 Houghton Mifflin Harcourt Publishing Compan Module Lesson

15 B g () = - + First, create a table of values for and g(). g () = Now graph the function along with the parent function Reflect 3. How is the graph of g () = - + related to the graph of the parent function ƒ () =?. In general, how is the graph of g () = - h + k related to the graph of ƒ () =? Houghton Mifflin Harcourt Publishing Compan YourTurn 5. Graph the absolute value function g () = along with the parent function f () = Module Lesson

16 Eplain Constructing Functions for Given Graphs of Absolute Value Functions You can write an absolute value function from a graph of the function. Eample Write an equation for each absolute value function whose graph is shown. A h is the number of units that the parent function is translated horizontall. For a translation to the right, h is positive; for a translation to the left, h is negative. In this situation, h =. k is the number of units that the parent function is translated verticall. For a translation up, k is positive; for a translation down, k is negative. In this situation, k = 3. The function is g () = B h is the number of units that the parent function is translated horizontall. For a translation to the right, h is positive; for a translation to the left, h is negative. In this situation, h =. k is the number of units that the parent function is translated verticall. For a translation up, k is positive; for a translation down, k is negative. In this situation, k =. The function is g () = - +. Houghton Mifflin Harcourt Publishing Compan Reflect 6. If the graph of an absolute value function is a translation of the graph of the parent function, eplain how ou can use the verte of the translated graph to help ou determine the equation for the function. Module Lesson

17 7. YourTurn Eplain 3 Graphing Stretches and Compressions of Absolute Value Functions You can compare the graphs of absolute value functions in the form g () = a, where a is a real number, with the graph of the parent function ƒ () = to see how a affects the absolute value function. Eample 3 Graph each absolute value function. A g () = - B g () = Houghton Mifflin Harcourt Publishing Compan Reflect. Describe how the graphs of g () = 1 and h () = - compare with the graph of ƒ () =. Use either the word stretch or shrink, and include the directions of movement. 9. What other transformation occurs when the value of a in g () = a is negative? Module Lesson

18 Your Turn Graph each absolute value function. 10. g () = g () = Elaborate 1. Wh is it important to note both the direction and the distance that a point has been translated either verticall or horizontall? 13. How does knowing a point in the graph other than the verte help ou find the value of a? 1. When graphing an absolute value function, how are g () = a and h () = - a related? 15. Essential Question Check-In How would the graph of the parent function ƒ () = be affected if h > 0, k < 0 and a > 1? Houghton Mifflin Harcourt Publishing Compan Module Lesson

19 Evaluate: Homework and Practice Graph each absolute value function. 1. g () = g () = - + Online Homework Hints and Help Etra Practice g () = g () = g () = g () = Houghton Mifflin Harcourt Publishing Compan Write an equation for each absolute value function whose graph is shown Module Lesson

20 Determine the domain and range of each function. 13. g () = g () = g () = g () = Graph each absolute value function. 17. g () = 3 1. g () = Houghton Mifflin Harcourt Publishing Compan Module Lesson

21 19. g () = 1_ 0. g () = - _ Identif the values of h, k, and a given each absolute value function. A. ƒ () = B. ƒ () = C. ƒ () = D. ƒ () = E. ƒ () = Bod Temperature The average bod temperature of a human is generall accepted to be 9.6 F. Complete the absolute value function below describing the difference d() in degrees Fahrenheit of the temperature of an individual human and the average temperature of a human. How is the graph of d() related to the graph of the parent function f() =? d() = - Houghton Mifflin Harcourt Publishing Compan 3. Population Statistics The average height of an American man is 69.3 inches. Complete the absolute value function below describing the difference d() in inches of the height of an individual American man and the average height of an American man. How is the graph of d() related to the graph of the parent function f() =? d() = - Module Lesson

22 H.O.T. Focus on Higher Order Thinking. Make a Prediction Complete the table and graph all the functions on the same coordinate plane. How do the graphs of ƒ () = a and g () = a compare? g () = 1_ 3 g () = 1_ 3 g () = - 1_ 3 g () = - 1_ Multiple Representations Write an equation for each absolute value function whose graph is shown Note: One point on this graph is (10, ). - - Note: One point on this graph is (6, -9). Houghton Mifflin Harcourt Publishing Compan Module Lesson

23 7. Represent Real-World Problems From his drivewa at point (, 6), Kerr is adjusting his rearview mirror before backing out onto the street. The farthest object behind him to the left he can see is a neighbor s mailbo at (-, 1). The farthest object behind him to the right he can see is a telephone pole at (0, 1). Create an absolute value function in the form ƒ () = a - h + k, with Kerr at the verte, to represent the boundaries of Kerr s visual field in the rearview mirror. Graph the function, and label Kerr, the mailbo, and the telephone pole Houghton Mifflin Harcourt Publishing Compan Blue Jean Images/Corbis Module Lesson

24 Lesson Performance Task Geese are initiall fling south in a V-shaped pattern that can be modeled b the absolute value function f () = a - h + k, where a represents the growth or shrinkage of the distance between geese, k represents the height change of the flock, and h represents a left or right shift in the flock. A. Graph the original flock function, g () =. State the function s domain and range in words B. While fling south, the flock encounters a jet stream and is forced to drop feet. Write the new equation and graph this function along with the original. State the new function s domain and range in words C. A short while after passing through the jet stream, the flock of geese encounters a rain storm and is forced to double the distance between each of its members in order to avoid colliding with one another. Write the new equation and graph all three functions. State the new function s domain and range in words Module Lesson

25 Name Class Date 13.3 Solving Absolute Value Equations Essential Question: How can ou solve an absolute value equation? Resource Locker Eplore Solving Absolute Value Equations Graphicall Absolute value equations differ from linear equations in that the ma have two solutions. This is indicated with a disjunction, a mathematical statement created b a connecting two other statements with the word or. To see wh there can be two solutions, ou can solve an absolute value equation using graphs. A B Solve the equation 5 =. Plot the function ƒ() = 5 on the grid. Then plot the function g() = as a horizontal line on the same grid, and mark the points where the graphs intersect. Write the solution to this equation as a disjunction: = or = Reflect 1. Wh might ou epect most absolute value equations to have two solutions? Wh not three or four? Houghton Mifflin Harcourt Publishing Compan. Is it possible for an absolute value equation to have no solutions? one solution? If so, what would each look like graphicall? Module Lesson 3

26 Eplain 1 Solving Absolute Value Equations Algebraicall To solve absolute value equations algebraicall, first isolate the absolute value epression on one side of the equation the same wa ou would isolate a variable. Then use the rule: If = a (where a is a positive number), then = a OR = a. Notice the use of a disjunction here in the rule for values of. You cannot know from the original equation whether the epression inside the absolute value bars is positive or negative, so ou must work through both possibilities to finish isolating. ]Eample 1 Solve each absolute value equation algebraicall. Graph the solutions on a number line. A 3 + = Subtract from both sides. 3 = Rewrite as two equations. 3 = 6 or 3 = 6 Solve for. = or = B = 19 Add to both sides. 3-5 = Divide both sides b = Rewrite as two equations. - 5 = or 5 = Add 5 to all four sides. = or = Solve for. = or = - _ Your Turn Solve each absolute value equation algebraicall. Graph the solutions on a number line _ + = = Houghton Mifflin Harcourt Publishing Compan Module Lesson 3

27 Eplain Absolute Value Equations with Fewer than Two Solutions You have seen that absolute value equations have two solutions when the isolated absolute value epression is equal to a positive number. When the absolute value is equal to zero, there is a single solution because zero is its own opposite. When the absolute value epression is equal to a negative number, there is no solution because absolute value is never negative. Eample Isolate the absolute value epression in each equation to determine if the equation can be solved. If so, finish the solution. If not, write no solution. A = 1 Subtract from both sides = 10 Divide both sides b 5. Absolute values are never negative. + 1 = No Solution 3_ B 3 = 3 5 Add 3 to both sides. 3_ = 5 Multipl both sides b _ 5. = 3 Rewrite as one equation. = Add to both sides. = Divide both sides b. = Houghton Mifflin Harcourt Publishing Compan Your Turn Isolate the absolute value epression in each equation to determine if the equation can be solved. If so, finish the solution. If not, write no solution _ = _ = 7 Module Lesson 3

28 Elaborate 7. Wh is important to solve both equations in the disjunction arising from an absolute value equation? Wh not just pick one and solve it, knowing the solution for the variable will work when plugged backed into the equation?. Discussion Discuss how the range of the absolute value function differs from the range of a linear function. Graphicall, how does this eplain wh a linear equation alwas has eactl one solution while an absolute value equation can have one, two, or no solutions? 9. Essential Question Check-In Describe, in our own words, the basic steps to solving absolute value equations and how man solutions to epect. Houghton Mifflin Harcourt Publishing Compan Module Lesson 3

29 Evaluate: Homework and Practice Solve the following absolute value equations b graphing = = 9 Online Homework Hints and Help Etra Practice =. 3_ ( ) + 3 = Houghton Mifflin Harcourt Publishing Compan Solve each absolute value equation algebraicall. Graph the solutions on a number line. 5. = _ 3 + = Module Lesson 3

30 = = Isolate the absolute value epressions in the following equations to determine if the can be solved. If so, find and graph the solution(s). If not, write no solution. 9. 1_ = = ( + + 3) = = Solve the absolute value equations = _ + 3 1_ - = Houghton Mifflin Harcourt Publishing Compan Module Lesson 3

31 15. ( + 5) = = The bottom of a river makes a V-shape that can be modeled with the absolute value function, d(h) = 1_ h 0, where d is the depth of the river bottom (in feet) and h is the horizontal 5 distance to the left-hand shore (in feet). A ship risks running aground if the bottom of its keel (its lowest point under the water) reaches down to the river bottom. Suppose ou are the harbormaster and ou want to place buos where the river bottom is 30 feet below the surface. How far from the left-hand shore should ou place the buos? Houghton Mifflin Harcourt Publishing Compan Image Credits: Motis/ Shutterstock 1. A flock of geese is fling past a photographer in a V-formation that can be described using the absolute value function b(d) = 3_ d 50, where b(d) is the distance (in feet) of a goose behind the leader, and d is the distance from the photographer. If the flock reaches 7 feet behind the leader on both sides, find the distance of the nearest goose to the photographer. Module Lesson 3

32 19. Geometr Find the points where a circle centered at (3, 0) with a radius of 5 crosses the -ais. Use an absolute value equation and the fact that all points on a circle are the same distance (the radius) from the center. 5 (3, 0) 0. Select the value or values of that satisf the equation 1_ = 1. A. = 5_ 3 B. = - 5_ 3 C. = 1_ 3 D. = - 1_ 3 E. = 3 F. = -3 G. = 1 H. = Terr is tring to place a satellite dish on the roof of his house at the recommended height of 30 feet. His house is 3 feet wide, and the height of the roof can be described b the function h() = 3_ 16 +, where is the distance along the width of the house. Where should Terr place the dish? H.O.T. Focus on Higher Order Thinking. Eplain the Error While attempting to solve the equation 3 - = 3, a student came up with the following results. Eplain the error and find the correct solution: = = 7 - = - 7_ 3 - = - 7_ 3 or - = 7_ 3 = 5_ 3 or = _ 19 3 Houghton Mifflin Harcourt Publishing Compan Image Credits: Patti McConville/Alam Module Lesson 3

33 3. Communicate Mathematical Ideas Solve this absolute value equation and eplain what algebraic properties make it possible to do so. 3 - = Justif Your Reasoning This absolute value equation has nested absolute values. Use our knowledge of solving absolute value equations to solve this equation. Justif the number of possible solutions = 10 Houghton Mifflin Harcourt Publishing Compan 5. Check for Reasonableness For what tpe of real-world quantities would the negative answer for an absolute value equation not make sense? Module Lesson 3

34 Lesson Performance Task A snowball comes apart as a child throws it north, resulting in two halves traveling awa from the child. The child is standing 1 feet south and 6 feet east of the school door, along an east-west wall. One fragment flies off to the northeast, moving feet east for ever 5 feet north of travel, and the other moves feet west for ever 5 feet north of travel. Write an absolute value function that describes the northward position, n(e), of both fragments as a function of how far east of the school door the are. How far apart are the fragments when the strike the wall? Houghton Mifflin Harcourt Publishing Compan Module Lesson 3

35 Name Class Date 13. Solving Absolute Value Inequalities Essential Question: What are two was to solve an absolute value inequalit? Resource Locker Eplore Visualizing the Solution Set of an Absolute Value Inequalit You know that when solving an absolute value equation, it s possible to get two solutions. Here, ou will eplore what happens when ou solve absolute value inequalities. A Determine whether each of the integers from -5 to 5 is a solution of the inequalit + < 5. Write es or no for each number in the table. If a number is a solution, plot it on the number line. Number = -5 = - Solution? = -3 = = -1 = 0 = 1 = = 3 = = 5 Houghton Mifflin Harcourt Publishing Compan B Determine whether each of the integers from -5 to 5 is a solution of the inequalit + > 5. Write es or no for each number in the table. If a number is a solution, plot it on the number line Number = -5 = - = -3 = - = -1 = 0 = 1 = Solution? = 3 = = 5 Module Lesson

36 State the solutions of the equation + = 5 and relate them to the solutions ou found for the inequalities in Steps A and B. If is an real number and not just an integer, graph the solutions of + < 5 and + > 5. Graph of all real solutions of + < 5: Graph of all real solutions of + > 5: Reflect 1. It s possible to describe the solutions of + < 5 and + > 5 using inequalities that don t involve absolute value. For instance, ou can write the solutions of + < 5 as > -3 and < 3. Notice that the word and is used because must be both greater than -3 and less than 3. How would ou write the solutions of + > 5? Eplain.. Describe the solutions of + 5 and + 5 using inequalities that don t involve absolute value. Eplain 1 Solving Absolute Value Inequalities Graphicall You can use a graph to solve an absolute value inequalit of the form ƒ () > g () or ƒ () < g (), where ƒ () is an absolute value function and g () is a constant function. Graph each function separatel on the same coordinate plane and determine the intervals on the -ais where one graph lies above or below the other. For ƒ () > g (), ou want to find the -values for which the graph ƒ () is above the graph of g (). For ƒ () < g (), ou want to find the -values for which the graph of ƒ () is below the graph of g (). Eample 1 A > Solve the inequalit graphicall. The inequalit is of the form ƒ () > g (), so determine the intervals on the -ais where the graph of ƒ () = lies above the graph of g () = Houghton Mifflin Harcourt Publishing Compan The graph of ƒ () = lies above the graph of g () = to the left of = -6 and to the right of = 0, so the solution of > is < -6 or > 0. - Module Lesson

37 B < 1 The inequalit is of the form f () < g (), so determine the intervals on the -ais where the graph of f () = lies the graph of g () = 1. The graph of f () = lies the graph of g () = 1 between = and =, so the solution of < 1 is > and < Reflect 3. Suppose the inequalit in Part A is instead of >. How does the solution change?. In Part B, what is another wa to write the solution > - and < 6? 5. Discussion Suppose the graph of an absolute value function ƒ () lies entirel above the graph of the constant function g (). What is the solution of the inequalit ƒ () > g ()? What is the solution of the inequalit ƒ () < g ()? Your Turn 6. Solve graphicall. Houghton Mifflin Harcourt Publishing Compan Module Lesson

38 Eplain Solving Absolute Value Inequalities Algebraicall To solve an absolute value inequalit algebraicall, start b isolating the absolute value epression. When the absolute value epression is b itself on one side of the inequalit, appl one of the following rules to finish solving the inequalit for the variable. Solving Absolute Value Inequalities Algebraicall 1. If > a where a is a positive number, then < -a or > a.. If < a where a is a positive number, then -a < < a. Eample A Solve the inequalit algebraicall. Graph the solution on a number line > 1 - > 6 B < -6 or - > 6 - < -10 or - > > 10 or < - The solution is > 10 or < -. + and + and The solution is and, or Reflect 7. In Part A, suppose the inequalit were > 1 instead of > 1. How would the solution change? Eplain.. In Part B, suppose the inequalit were instead of How would the solution change? Eplain. Houghton Mifflin Harcourt Publishing Compan Module Lesson

39 Your Turn Solve the inequalit algebraicall. Graph the solution on a number line < Eplain 3 Solving a Real-World Problem with Absolute Value Inequalities Absolute value inequalities are often used to model real-world situations involving a margin of error or tolerance. Tolerance is the allowable amount of variation in a quantit. Eample 3 A machine at a lumber mill cuts boards that are 3.5 meters long. It is acceptable for the length to differ from this value b at most 0.0 meters. Write and solve an absolute value inequalit to find the range of acceptable lengths. Houghton Mifflin Harcourt Publishing Compan Analze Information Identif the important information. The boards being cut are meters long. The length can differ b at most 0.0 meters. Formulate a Plan Let the length of a board be l. Since the sign of the difference between l and 3.5 doesn t matter, take the absolute value of the difference. Since the absolute value of the difference can be at most 0.0, the inequalit that models the situation is l -. Solve l l and l l and l So, the range of acceptable lengths is l. Module Lesson

40 Justif and Evaluate The bounds of the range are positive and close to, so this is a reasonable answer. The answer is correct since = 3.5 and = 3.5. Your Turn 11. A bo of cereal is supposed to weigh 13. oz, but it s acceptable for the weight to var as much as 0.1 oz. Write and solve an absolute value inequalit to find the range of acceptable weights. Elaborate 1. Describe the values of that satisf the inequalities < a and > a where a is a positive constant. 13. How do ou algebraicall solve an absolute value inequalit? 1. Eplain wh the solution of > a is all real numbers if a is a negative number. 15. Essential Question Check-In How do ou solve an absolute value inequalit graphicall? Houghton Mifflin Harcourt Publishing Compan Module Lesson

41 Evaluate: Homework and Practice 1. Determine whether each of the integers from -5 to 5 is a solution of the inequalit If a number is a solution, plot it on the number line Online Homework Hints and Help Etra Practice Determine whether each of the integers from -5 to 5 is a solution of the inequalit If a number is a solution, plot it on the number line Solve each inequalit graphicall > Houghton Mifflin Harcourt Publishing Compan 5. 1_ + < Module Lesson

42 Match each graph with the corresponding absolute value inequalit. Then give the solution of the inequalit. A. + 1 > 3 B. + 1 < 3 C. - 1 > 3 D. - 1 < Solve each absolute value inequalit algebraicall. Graph the solution on a number line _ + 3 > Houghton Mifflin Harcourt Publishing Compan Module Lesson

43 < < Houghton Mifflin Harcourt Publishing Compan < Module Lesson

44 Solve each problem using an absolute value inequalit. 17. The thermostat for a house is set to 6 F, but the actual temperature ma var b as much as F. What is the range of possible temperatures? 1. The balance of Jason s checking account is $30. The balance varies b as much as $0 each week. What are the possible balances of Jason s account? 19. On average, a squirrel lives to be 6.5 ears old. The lifespan of a squirrel ma var b as much as 1.5 ears. What is the range of ages that a squirrel lives? Houghton Mifflin Harcourt Publishing Compan Image Credits: (t) Burwell and Burwell Photograph/iStockPhoto.com; (b) Flickr/Nanc Rose/Gett Images Module Lesson

45 0. You are plaing a histor quiz game where ou must give the ears of historical events. In order to score an points at all for a question about the ear in which a man first stepped on the moon, our answer must be no more than 3 ears awa from the correct answer, What is the range of answers that allow ou to score points? 1. The speed limit on a road is 30 miles per hour. Drivers on this road tpicall var their speed around the limit b as much as 5 miles per hour. What is the range of tpical speeds on this road? Houghton Mifflin Harcourt Publishing Compan Image Credits: (t) NASA Headquarters - Great Images in NASA (NASA-HQ-GRIN); (b) Image Source/ Gett Images Module Lesson

46 H.O.T. Focus on Higher Order Thinking. Represent Real-World Problems A poll of likel voters shows that the incumbent will get 51% of the vote in an upcoming election. Based on the number of voters polled, the results of the poll could be off b as much as 3 percentage points. What does this mean for the incumbent? 3. Eplain the Error A student solved the inequalit > 1 graphicall. Identif and correct the student s error. I graphed the functions () = and g() = 1. Because the graph of g() lies above the graph of () between = -3 and = 5, the solution of the inequalit is -3 < < Houghton Mifflin Harcourt Publishing Compan Module 13 6 Lesson

47 . Multi-Step Recall that a literal equation or inequalit is one in which the constants have been replaced b letters. a. Solve a + b > c for. Write the solution in terms of a, b, and c. Assume that a > 0 and c 0. b. Use the solution of the literal inequalit to find the solution of > 1. c. Eplain wh ou must assume that a > 0 and c 0 before ou begin solving the literal inequalit. Houghton Mifflin Harcourt Publishing Compan Module Lesson

48 Lesson Performance Task The distance between the Sun and each planet in our solar sstem varies because the planets travel in elliptical orbits around the Sun. Here is a table of the average distance and the variation in the distance for the five innermost planets in our solar sstem. Average Distance Variation Mercur 36 million miles 7.5 million miles Venus 67. million miles 0.5 million miles Earth 9.75 million miles 1.75 million miles Mars 11 million miles 13 million miles Jupiter million miles million miles a. Write and solve an inequalit to represent the range of distances that can occur between the Sun and each planet. b. Calculate the percentage variation (variation divided b average distance) in the orbit of each of the planets. Based on these percentages, which planet has the most elliptical orbit? Houghton Mifflin Harcourt Publishing Compan Image Credits: JPL/NASA Module 13 6 Lesson