Lesson 3-7: Absolute Value Equations Name:
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1 Lesson 3-7: Absolute Value Equations Name: In this activity, we will learn to solve absolute value equations. An absolute value equation is any equation that contains an absolute value symbol. To start, let s review a little of what we know about the absolute value function. 1. a. 2 b. 2 c. 3 d Circle the numbers in the set S that can be substituted for x to make the equation x 2 true. S 3, 2, 1, 0, 1, 2, 3 3. Circle the numbers in the set S that can be substituted for y to make the equation y 1 true. S 3, 2, 1, 0, 1, 2, 3 Almost every absolute value equation has two answers: In the equation 6, x could be equal to 6, or x could be equal to 6. Either value will make the equation true:, and Use the principle above to fill in the blanks for each question below: 4. x 5 means that x could be equal to or x could be equal to. 5. x 13 means that x could be equal to or x could be equal to. 6. x 250 means that x could be equal to or x could be equal to. Every absolute value equation represents two equations combined into one: 10 means that x = 10 or x = -10. means that or Use this principle to fill in the blanks for each question below: 7. x 3 5 means x 3 5 or 8. 2x 1 9 means or 9. 53x 25 means or x 10 changes to which means or 11. 3e 8 7 changes to which means or Algebra 1 OBJ: Introduce solving absolute value equations Skill A
2 Skill 3: Solve Absolute Value Equations To solve absolute value equations, solve the two equations each represents: means or y = 6 or y = -2 The solution set is y = 6 or y = -2 Check: Does? Does? Yes. Yes. Use this principle to solve each absolute value equation below and check the solutions. 12. x x x x e Scrambled answers for #12-16: -8, - 3, -5, -4, -1, 1, 2, 5, 5, 10 Algebra 1 OBJ: Introduce solving absolute value equations Skill 4 3-7B
3 Answer each question below: More 3-7: Absolutely Less 1. True or False? a. 1 < 2 b. -3 < 2 c. -1 < 2 d. 0 < 2 2. Circle numbers in the set S that can be substituted for x to make the inequality x < 2 true. S 3, 2.75, 2, 1.33, 1, 0, 0.5, 1, 1.99, 2, 2.66, 3 3. Circle numbers in the set S that can be substituted for y to make the equation y < 2 true. S 3, 2.75, 2, 1.33, 1, 0, 0.5, 1, 1.99, 2, 2.66, 3 Every absolute value inequality with < or represents two inequalities combined with and : x < 2 means that x < 2 and x > -2. It means x < 2 because for numbers less than 2, the absolute value will be less than 2: 1 < 2 It means x > -2 because for numbers greater than 2, the absolute value will be less than 2: -1 < 2 The numbers that work in x < 2 must meet both of these requirements: -1.7 works because it is less than 2 and greater than 2, so -1.7 < 2 is a true statement. -3 does not work, even though it is less than 2, because it is not greater than 2, so -3 < 2 is not true. Use the principle above to fill in the blanks for each question. 4. x 5 means x 5 and 5. x < 250 means and 6. x + 3 < 5 means and 7. 2x 1 9 changes to which means and Skill 4: Solve absolute value inequalities To solve absolute value inequalities, solve the two inequalities that each represents. For example, to solve y - 2 4: y means y 2 4 and y 6 and y -2 The solution set is y 6 and y -2, which is the same as. The graph of this solution set looks like: Check x = 5: Is 5-2 4? Check x = -3: Is ? 3 4 Yes No. (x = 5 is in solution set.) (x = -3 is not in solution set.) Algebra 1 OBJ: Introduce solving absolute value inequalities Skill C
4 Use this principle to solve each absolute value inequality, graph the solution set, and check two values one in the solution set and one not in the solution set. 8. 2x y 8 < y e 2 < x + 4 < (w 1) 9 Algebra 1 OBJ: Introduce solving absolute value inequalities Skill D
5 Even More 3-7: More Absolutely Every absolute value inequality with > or represents two inequalities combined with or : x 4 means that x 4 or x -4. It means x 4 because for numbers greater than 4, the absolute value will be greater than 4: 6.3 > 4 It means x 4 because for numbers less than 4, the absolute value will be greater than 4: 8 > 4 The numbers that work in x 4 can meet either of these requirements, but don t have to meet both: 9 works because it is greater than 4, even though it is not less than is a true statement works, even though it is not greater than 4, because it is less than is a true statement. Use the principle above to fill in the blanks for each question below: 1. x 1 > 7 means x 1 7 or x > 25 means or 3. 3e 8 > 7 means or 4. 4y 8 > 16 changes to which means or Skill 4: Solve absolute value inequalities To solve absolute value inequalities, solve the two inequalities that each represents. For example, to solve 2x + 1 > 5 : 2x + 1 > 5 means 2x + 1 > 5 or 2x + 1 < x 4 x 2 or The solution set is x > 2 or x < 3. The graph of this solution set looks like: 2x -6 x Check x = 4: Is 2( 4) + 1 > 5? Check x = 0: Is 2(0) + 1 > 5? 7 > 5 Yes. 1 > 5 No. ( 4 is in solution set.) (0 is not in solution set.) Use the principle above to solve each absolute value equation, graph the solution set, and check two values one in the solution set and one not in the solution set. 5. w Algebra 1 OBJ: Expand solving absolute value inequalities Skill E
6 6. t + 4 > y x > z y > Compare the process of solving inequalities with < or to solving inequalities with > or. What is the same? What s different? Algebra 1 OBJ: Expand solving absolute value inequalities Skill F
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