ࡱ> qsnopx (bjbj 2w Y Y YYYY8UY4ZY\___M`   LNNNNNN3vN Y j   N_M`ѦG!!! X)_^ YM`L! L!!d(H¶M` 5 Y 8]00K| 6K¶K Yжh  !     NN&       K          : Absolute Value Equations and Inequalities Reporting Category Equations and Inequalities Topic Solving absolute value equations and inequalities Primary SOL AII.4a The student will solve, algebraically and graphically, absolute value equations and inequalities. Graphing calculators will be used for solving and for confirming the algebraic solutions. Related SOL AII.6 Materials Graphing calculators Activity 1 Cards (attached) Three attached handouts Vocabulary union, intersection, linear inequality, absolute value, distance, linear equation (earlier grades) compound inequality, absolute value inequality (AII.4a) Student/Teacher Actions (what students and teachers should be doing to facilitate learning) Invoke students prior learning by asking questions about absolute value. Use questions such as the following: What is absolute value? What does the graph of the absolute value of a linear function look like? What are the solutions to |x| = 5? What are some solutions to |x| < 5? What are some solutions to |x 2| e" 4? Explain to students that the objective of the lesson is to learn three methods for solving absolute value equations and inequalities, which will give them greater insight into the solution process. Method 1: Graphing Ask students to graph  EMBED Equation.3  and  EMBED Equation.3  by hand and then on their graphing calculators. Next, ask students how they might use the graphs they generated to solve the following equation and inequalities: a.  EMBED Equation.3  b.  EMBED Equation.3  c.  EMBED Equation.3  Have students solve each one, using the graphs. Distribute copies of the attached Solve by Graphing handout, and have students complete it, using the graphing method. Method 2: Absolute Value as Distance Discuss the statement, Absolute value represents distance. Ask students whether this is true, and if so, what it means. Have the class discuss  EMBED Equation.3  in terms of distance. Ask what values make  EMBED Equation.3  true, and discuss this in terms of distance. Summarize responses by saying, We can describe the solution set as the set of points whose distance from the origin is equal to 3. Show the solution set on a number line.  Ask what changes in the solution set when we want to solve  EMBED Equation.3 . Summarize responses by saying, We can describe the solution set as the set of points whose distance from 1, rather than 0, is equal to 3. It is important to show the solution set on a number line. Ask, If the solution to  EMBED Equation.3  is the set of points whose distance from the origin is equal to 3, then what is the solution set to  EMBED Equation.3 ? Discuss this question, asking students to identify some values of x that make  EMBED Equation.3  true. They should come to see that the solution set is the set of points whose distance from the origin is less than 3. Show the solution set on a number line.  Ask how the graph of this solution set would be different if you were solving  EMBED Equation.3 . If you were solving EMBED Equation.3 . If you were solving  EMBED Equation.3 . Instruct students to use the distance method to solve the following examples: a.  EMBED Equation.3  b.  EMBED Equation.3  c.  EMBED Equation.3  d.  EMBED Equation.3  Then, have them state the solution of each example, using the sentence frame in the box below. Finally, have them graph each solution set on a number line.  SHAPE \* MERGEFORMAT  Discuss the validity of  EMBED Equation.3 . Ask whether it is true. Ask whether students can think of an example in which it is not true. Once its validity is established, show this example:  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Show on a number line that the solution set is the set of points whose distance from  EMBED Equation.3 is less than  EMBED Equation.3 . Therefore, we can write the solution set in set-builder notation as  EMBED Equation.3 . Method 3: Absolute Value as Compound Inequality Explain the method of writing absolute value equations and inequalities as compound statements. Students should make the connection quickly if this is the third method discussed.  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  OR  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  AND  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  OR  EMBED Equation.3  Have students write the following as compound statements, solve each branch, and graph the solution set to each:  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Give every pair of students a set of Activity 1 Cards. Explain that student pairs will play a game to match an inequality with the graph of its solution set, the statement describing its solution set, and the corresponding compound statement. Have pairs shuffle their cards and deal 8 cards each. In turn, each player places one card on the table. If there is a corresponding card already on the table, the player places his/her card on top of that card to create a stack; if not, the player starts a new stack. At the end of the game, six stacks should have been created. When a student places the fourth card on any stack, that student collects that stack. The player with the most stacks wins. Once the game becomes too repetitive with these cards, have students create their own game cards in the same manner by writing inequalities, accompanying statements describing the solution sets, accompanying graphs, and accompanying compound inequalities. Distribute copies of the attached Activity 2 Problems, and have students work in pairs to come up with three equations or inequalities that the provided graphs can help them solve. Have them also write the solutions. The Activity 3 Problems are designed as a review and practice session on these topics, with students assisting each other through the review process. Set up six workstations with each station containing multiple sheets showing one set of problems. Be sure that each station has enough sheets for every student to have one. Place a poster at each station with the answers and full solutions to the problems. Put students into groups at the six stations. Give each group 7 to 10 minutes per station to complete the problems found there, working together and checking answers. Set a timer so that the groups know when to stop working and rotate to the next station. Assessment Questions You have noticed when you graph certain absolute value inequalities that the solution set is everything within an interval between two points, as demonstrated below:  In other absolute value inequalities, the solution set is everything on the number line outside of an interval, as demonstrated below: What types of absolute value inequalities lead to each of these kinds of solution sets? Why? Given any interval between two given numbers, how can you write an absolute value inequality to produce that interval as its solution set? Journal/Writing Prompts Discuss the three ways of solving absolute value equations and inequalities, including which one you think makes the most sense and why. Explain the steps you would use to solve an absolute value inequality, using your method of choice. Write a fictitious story about the history of absolute value. Extensions and Connections (for all students) Have students solve problems where the argument of the absolute value is quadratic. Have students work in groups to reteach one of the methods, and record a video for student review. Have students develop their own mnemonic to remember the connecting compound word, and or or, for each absolute value equation or inequality type. Have students solve  EMBED Equation.3 , using a distance argument. Strategies for Differentiation When teaching the compound statement method, consider using a template like the one shown at right. Provide additional sentence frames to help students articulate solution sets to absolute value inequalities. Restrict the number of game cards to two representations for students who struggle with multiple representations. Construct a number line on the floor, and have students physically represent distances from zero and solution sets to absolute value equations and inequalities. Allow students to use talking graphing calculators as they interpret graphs and points of intersection. Solve by Graphing Solve by graphing each of the following on a number line: 1. 3 < x d" 5 2. x < "1 or x e" 5 3. x e" 0 and x < 4 4. x e" 4 and x < 7 5. x < "5 and x > 1 6. x e" 4 and x > 7 7. x e" 4 or x > 7 8. x > "1 or x < 3 Graph on a number line the real numbers that satisfy each of the following: 9. |x| = 5 10. |x| = 0 11. |x| d" 8 12. |x| e" 6 13. |x| e" "1 14. |x| d" 0 15. |x " 3| d" 5 16. |x + 3| d" 6 17. |x " 4| e" 3 18. |2x " 5| d" 16 Activity 1 Cards Copy cards on cardstock, and cut apart on the dotted lines. |x + 5| d" 4The set of points whose distance from "5 is less than or equal to 4x + 5 e" "4 and x + 5 d" 4The set of points whose distance from 5 is less than 4|x " 5| < 4x " 5 > "4 and x " 5 < 4|x + 4| e" 5The set of points whose distance from "4 is greater than or equal to 5x + 4 d" "5 or x + 4 e" 5|2x " 1| d" 7The set of points whose distance from  EQ \F(1,2) is less than or equal to  EQ \F(7,2)2x " 1 e" "7 and 2x " 1 d" 7|2x + 1| e" 7The set of points whose distance from " EQ \F(1,2) is greater than or equal to  EQ \F(7,2)2x + 1 d" "7 or 2x + 1 e" 7|x " 2| d" 7The set of points whose distance from 2 is less than or equal to 7x " 2 e" "7 and x " 2 d" 7 Activity 2 Problems For each problem, write an absolute value equation and two inequalities that can be solved using the given graph. Then, solve, write the solution set in set-builder notation, and graph the solution set on a number line. 1. Absolute value equationInequality 1Inequality 2Solution set in set-builder notationSolution set in set-builder notationSolution set in set-builder notationGraph of solution setGraph of solution setGraph of solution set  2. Absolute value equationInequality 1Inequality 2Solution set in set-builder notationSolution set in set-builder notationSolution set in set-builder notationGraph of solution setGraph of solution setGraph of solution set Activity 3 Problems Station 1 Problems Solve and graph the following: 1. 3(x " 2) e" 5 2. 6 " 4x d" 10 3.  EQ  EQ \F(x,3)  + 5 d" 7 4. "5 d" 2x + 1 d" 3 5. 7x " (3 " 2x) e" x " 3 Station 2 Problems Solve and graph the following: 1. 3x + 6 e" 4 or 6x " 2 e" 8 2. 4x + 2 e" 10 or "4x " 2 e" 10 3.  EQ \F(x, 4)  + 6 > 10 and  EQ \F(2x  1,3)  > 2 4. "5x < 10 or 4x < 16 5. Graph: x > 5 and x > 9 Station 3 Problems Solve and graph the following: 1. |2x + 5| = 3 2. |5 " x| + 4 = 10 3. "2|x + 6| = "10 4. |2x " 3| d" 5 5. |11 " 3x| + 6 > 10 Station 4 Problems Solve and graph the following: 1. 3|2x " 2| + 8 = 23 2. |3x " 5| e" 4 3. 4|"x + 4| ( 12 4. |5 " (2x + 3)| e" 4 5. |2 " x| ( 9 Station 5 Problems Write an inequality for each of the following: You have $20,000 available to invest in stocks A and B. 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The largest egg of any bird is that of the ostrich. An ostrich egg can reach 8 inches in length. The smallest egg is that of the hummingbird. Its eggs are approximately 0.4 inches in length. Write an inequality that represents the range of lengths of bird eggs. On Pennsylvanias interstate highway, the speed limit is 55 mph. The minimum speed limit is 45 mph. Write a compound inequality that represents the allowable speeds. You have $60 to spend and a coupon that allows you to take $10 off any purchase of $50 or more at the department store. Write an inequality that describes the possible retail value of the items you can buy if you use this coupon. Write an inequality that describes the different amounts of money you can spend if you use the coupon. Develop a context for the inequality 0 < 2x " 5 < 30. Develop a context for the inequality |x " 7| > 12. Station 6 Problems Set up and solve each of the following inequalities: Maria has grades of 79, 67, 83, and 90 on four tests. 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