1-7. Solving Absolute Value Equations. Going Deeper. Standards for Mathematical Content Give students the equation _3 x + 1 = 4. Have.

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1 -7 Solving Absolute Value Equations Going Deeper Essential question: How can ou use graphing to solve equations involving absolute value? INTRODUCE Standards for Mathematical Content Give students the equation _3 + =. Have 3 + and =. Ask students students graph = _ to find the intersection of the graphs, which is at (, ). Have students solve the equation algebraicall as well. Point out to students that the solution,, is the -coordinate of the intersection point. A-CED.. Create equations in one variable and use them to solve problems.* A-CED.. Create equations in two variables to represent relationships between quantities; graph equations on coordinate aes with labels and scales.* A-REI.. Eplain each step in solving a simple equation as following from the equalit of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. A-REI.. Eplain wh the -coordinates of the points where the graphs of the equations = f () and = g() intersect are the solutions of the equation f () = g(); Include cases where f () and/or g() are linear, absolute value functions.* Also: N-Q..*, N-Q..*, F-IF.., F-IF..*, F-IF.3.7*, F-IF.3.7b* TEACH Questioning Strategies In part C, wh do ou identif the -coordinate of the intersection point as the solution instead of the -coordinate? It is the -coordinate that satisfies the original equation. Wh is = 5 separated into two functions in part A and part B? To solve an Solving linear equations equation graphicall, ou graph each side as a separate equation and then find the -coordinates of an intersection points. Prerequisites etra eample Solve the equation = 7 b graphing. 0, Math Background To solve an absolute value equation, a - h + k = c, first graph the equation = a - h + k. Then, graph the horizontal line = c. Estimate the -coordinates of the points of intersection. There ma be 0,, or solutions. The eact solutions are found algebraicall b isolating the absolute value epression on the left and then setting the epression inside the absolute value bars equal to the number on the right and its opposite. So for + =, the solutions are found b solving + = and + = -. Chapter Lesson 7

2 Name Class Date -7 Notes Solving Absolute Value Equations Going Deeper Essential question: How can ou use graphing to solve equations involving absolute value? The equation = 5 is an eample of an absolute value equation. A-REI.. Solving an Absolute Value Equation b Graphing Use a graphing calculator to solve the equation = 5. A Treat the left side of the equation as the absolute value equation = Treat the right side as the equation = 5. B Press Y=. Enter the first equation, = - 3 +, as Y and the second equation = 5 as Y. C Press GRAPH. Cop the graph from our calculator on the grid at the right. Use the intersect feature under the CALC menu to find the points of intersection of Y and Y D Identif the -coordinate of each point where the graphs of Y and Y intersect. Show that each -coordinate is a solution of = 5. and 5; = - + = + = 5; = + = + = 5 REFLECT a. Wh is the -coordinate of both points of intersection equal to 5? All points that satisf the equation = 5 have a -coordinate of 5. Since the points of intersection must satisf both equations, the both must have a -coordinate of 5. b. The verte of an absolute value graph is the lowest point if the graph opens upward or the highest point if the graph opens downward. The verte of the graph of = is (, 3). How are the coordinates of the verte related to its equation? The -coordinate of the verte is the number after the + sign and the -coordinate of the verte is the number after the - sign. In general, if the equation is = a - b + c, the verte is (c, b). Chapter Lesson 7 A-REI.. Solving an Absolute Value Equation Using Algebra Solve the equation = 5 using algebra. A Isolate the epression = 5 Write the equation. - - Subtract from both sides. B C - 3 = Simplif. - 3 = Divide both sides b. - 3 = Simplif. Interpret the equation - 3 = : What numbers have an absolute value equal to? and - Set the epression inside the absolute value bars equal to each of the numbers from Part B and solve for. - 3 = or - 3 = - Write an equation for each value of Add 3 to both sides of each equation. REFLECT = 5 or = Simplif. a. The left side of the equation is Evaluate this epression for each solution of the equation. How does this help ou check the solutions? When = 5, = + = + = 5; when =, = - + = + = 5; the value of the epression is 5 in each case, and this is the number on the right side of the equation, so the solutions check. b. Suppose the number on the right side of the equation was -5 instead of 5. What solutions would the equation have? Wh? When answering these questions, ou ma want to refer to the graph of = There would be no solutions because the graph of = -5 does not intersect the graph of = Chapter Lesson 7 Chapter Lesson 7

3 Questioning Strategies Can ou predict the number of solutions to the equation a h + k = c before solving? If c = k, the equation becomes a - h = 0, and ou can predict there will be solution, h. If c k, ou can predict there will be either 0 or solutions. EXTRA Solve the equation = 7 using algebra. 0, 3 Questioning Strategies Wh is (0, ) a point of the graph showing Sal s distance from home? After 0 minutes, Sal is 0 min 0. mi/min = miles from home. Since the horizontal ais represents time and the vertical ais represents distance, (0, ) is on the graph. Wh is the verte of the graph important? It is the point at which Sal turns around and starts the return home. Avoid Common Errors Make sure students understand that the -coordinate shows the distance from home, not the distance traveled. EXTRA During driving class, Harr drives directl to a point miles awa and then reverses his direction to drive back to school. He drives at a constant speed of 0.5 mile per minute. Write and graph a model that gives his distance d (in miles) from school as a function of the elapsed time t (minutes). Use the graph to find the time(s) when Harr is miles from school. d = -0.5 t - + ; t = min; t = min Distance (miles) CLOSE 5 d 3 0 Time (minutes) Essential Question How can ou use graphing to solve equations involving absolute value? Graph two equations, each representing the epression on one side of the equal sign. Find the points where the graphs intersect. The -coordinates of those points are the solutions of the original equation. t Highlighting the Standards 3 addresses Mathematical Practice Standard (Model with mathematics). Both the graph and the equation = -0. t represent Sal s distance from home at a given time. To find when Sal is a particular distance from home ( mile in the case of the Eample) ou can solve the equation either graphicall, as in Part C, or algebraicall, as in Reflect Question 3a. Summarize Have students write a journal entr in which the describe how to solve c = a - h + k b graphing. PRACTICE Where skills are taught Where skills are practiced EXS. 3 EXS. 3 EXS. 7 Chapter 3 Lesson 7

4 3 A-CED.. Solving a Real-World Problem Notes Sal eercises b running east 3 miles along a road in front of his home and then reversing his direction to return home. He runs at a constant speed of 0. mile per minute. Write and graph an absolute value equation that gives his distance d (in miles) from home in terms of the elapsed time t (in minutes). Use the graph to find the time(s) at which Sal is mile from home. A Determine the three ke values of the distance equation: When Sal begins his run (t = 0 minutes), he is 0 miles from home, so d = 0. When Sal reverses direction, he is 3 miles from home. 3 miles He reaches this point in t = = 30 minutes, so 0. mile per minute when t = 30, d = 3. When Sal returns home, he is 0 miles from home. Because he has miles run a total of miles, he reaches this point in t = 0. mile per minute = 0 minutes, so when t = 0, d = 0. B Add ais labels and scales to the coordinate plane shown, then plot the points (t, d) using the time and distance values from part A. The equation is an absolute value equation, and the verte of the equation s graph is the point that represents when Sal reverses direction. Draw the complete graph and then write the absolute value equation. d = -0. t C To find the time(s) when Sal is mile from home, draw the graph of =. Find the t-coordinate of each point where the two graphs intersect. t = 0 minutes and t = 50 minutes REFLECT 3a. Show how to use algebra to find the time(s) when Sal is mile from home. 3 0 d Times (minutes) -0. t = ; -0. t - 30 = -; t - 30 = 0; t - 30 = 0 or t - 30 = -0; t = 50 or t = 0 Distance (miles) t Chapter 3 Lesson 7 PRACTICE Use a graphing calculator to solve each absolute value equation. Sketch our graphs on the grids provided = = = = -5 or = Solve each absolute value equation using algebra = = = = 0 or = - 7. The number of gallons of water in a storage tank is given b = -0 h where h is the time in hours since the tank was last empt. a. What is the maimum number of gallons the tank can hold? b. For what values of h is the tank half empt?. The number of shoppers in a store is modeled b = -0.5 t - + where t is the time (in minutes) since the store opened at 0:00 a.m. a. For what values of t are there 00 shoppers in the store? b. At what times are there 00 shoppers in the store? c. What is the greatest number of shoppers in the store? d. At what time does the greatest number of shoppers occur? = - or = 7 = -9 or = c. For what values of h is the tank empt? 0 and 0 d. if the tank is empt, how long does it take to refill it? 0 hours 0 and gallons 00 and 37 :0 P.M. and : P.M. : P.M = - No solution Chapter Lesson 7 Chapter Lesson 7

5 ADDITIONAL PRACTICE AND PROBLEM SOLVING Assign these pages to help our students practice and appl important lesson concepts. For additional eercises, see the Student Edition. Answers Additional Practice. {, }. { _, _ } 3. { 0, 0 }. { 9, 9 } 5. {, }. { 3, 7 } 7. {, 3 }. {, } 9. {, 0 } 0. { 3, 3 }. { }. { 3 } 3. two. one 5. none. = ; 7.5 Problem Solving. 70 = 0.0; 9.9 cm; 70.0 cm. 53 = 0.0; m; 53.0 m 3. and..9 cm 5. B. G 7. C Chapter 5 Lesson 7

6 Name Class Date Additional Practice -7 Notes Chapter 5 Lesson 7 Problem Solving Chapter Lesson 7 Chapter Lesson 7