New Jersey Center for Teaching and Learning. Progressive Mathematics Initiative

Transcription

1 Slide 1 / 70 New Jersey Center for Teaching and Learning Progressive Mathematics Initiative This material is made freely available at and is intended for the non-commercial use of students and teachers. These materials may not be used for any commercial purpose without the written permission of the owners. NJCTL maintains its website for the convenience of teachers who wish to make their work available to other teachers, participate in a virtual professional learning community, and/or provide access to course materials to parents, students and others. Click to go to website:

2 Slide 2 / 70 Graphing & Solving Absolute Values

3 Slide 3 / 70 Table of Contents Review Click on the topic to go to that section Equations Inequalities Functions

4 Slide 4 / 70 Review Return to Table of Contents

5 Slide 5 / 70 Remember... Absolute value is a number's distance from zero. Since it is a representation of distance, it cannot be less than zero. The symbol for absolute value is # Do not mistake the absolute value symbol of -652 with the grouping symbol of (-652).

6 Slide 6 / 70 Absolute value bars are grouping symbols so you follow the order of operations. Simplify within the absolute value bars before finding the absolute value. (-3)(15)

7 Slide 7 / 70 Examples (-3)(7) - -2 (-4) (-3) (2)(-4) (-8)

8 Slide 8 / 70 Try these (-2) (5)(9-4) (-3)(-2) + -3 (5)(-5)

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11 3 (4)(6-8) Slide 11 / 70

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13 5 - -2(3)(7) Slide 13 / 70

14 Slide 14 / 70 6 If r = 2 and s = -7, what is the value of r - s? A 5 B -5 C 9 D -9 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 2011.

15 Slide 15 / 70 Equations Return to Table of Contents

16 Slide 16 / 70 Number of Solutions Since absolute value indicates a number that is not negative, there are two answers to x = 3. They are -3 and 3 since both numbers are 3 spaces away from zero on a number line. We will use this knowledge to solve absolute value equations.

17 Slide 17 / 70 Absolute value equations have two, one or no solution depending on the value of the solution. If the equation's solution is positive, there are 2 solutions. x = 7 x = -7 or x = 7 If the equation's solution is zero, there is 1 solution. x = 0 x = 0 If the equation's solution is negative, there are 0 solutions. x = -2 No solution

18 Slide 18 / 70 7 How many solutions does the equation have? x = 17

19 Slide 19 / 70 8 How many solutions does the equation have? 5x - 8 = -3

20 Slide 20 / 70 9 How many solutions does the equation have? x + 13 = 0

21 Slide 21 / How many solutions does the equation have? -3x + 1 = 19

22 Slide 22 / 70 Solving Equations To solve an absolute value equation: Isolate the absolute value expression. Create two equations by setting the expression equal to the solution and its opposite. Solve the two separate equations.

23 Slide 23 / 70 Example: #1 x - 5 = 12 Isolate Absolute Value Expression x - 5 = 12 or x - 5 = -12 Create 2 equations by setting the expression equal to the solution and its opposite. x = 17 or x = -7 Solve the two equations

24 Slide 24 / 70 Example: #2 3x = x + 2 = 10 Isolate Absolute Value Expression 3x + 2 = 10 or 3x + 2 = -10 Create 2 equations by setting the expression equal to the solution and its opposite. 3x = 8 or 3x = -12 Solve the two equations x = or x =

25 Slide 25 / 70 Try These: 3x - 7 = 8 x = 11 5x + 1 = 0

26 Slide 26 / Solve x = 17

27 Slide 27 / Solve 5x - 8 = -3

28 Slide 28 / Solve x + 13 = 0

29 Slide 29 / Solve -3x + 1 = 19

30 Slide 30 / Solve -3x = 17

31 Slide 31 / 70 Writing Equations Number lines can be used to write absolute value equations when you are given two numbers as its solution. Graph the two numbers on a number line and find the midpoint. Use the midpoint and the points' distance from it to write the equation.

32 Slide 32 / 70 Example: Write an absolute value equation that has -2 and 8 as its solutions. 5 5 { { The solution is 5 points from the midpoint 3. The equation is written: x - midpoint = distance x - 3 = 5

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34 Slide 34 / Write an absolute value equation that has -3 and 5 as its solutions. A x - 1 = 4 B x - 4 = 1 C x + 1 = 4 D x + 4 = 1

35 Slide 35 / Write an absolute value equation that has 4 and 16 as its solutions. A x - 10 = 6 B x - 6 = 10 C x + 10 = -6 D x + 6 = 10

36 Slide 36 / Write an absolute value equation that has 0 and 5 as its solutions. A x + 5 = 2.5 B x = 2.5 C x = 5 D x = 2.5

37 Slide 37 / 70 Inequalities Return to Table of Contents

38 Slide 38 / 70 Absolute Value Inequalities There are two forms of absolute value inequalities. One with less than, a < b, and the other with greater than, a > b. They are solved differently. When an inequality contains an absolute value expression, it can be written as a compound inequality.

39 Slide 39 / 70 x < 6 means x < 6 and x > -6 x > 6 means x > 6 or x < -6

40 Slide 40 / What word would you use when writing the compound inequality for this equation? x - 7 > 15 A B or and

41 Slide 41 / What word would you use when writing the compound inequality for this equation? 3 - x < 4 A B or and

42 Slide 42 / What word would you use when writing the compound inequality for this equation? 9x < -1 A B or and

43 Slide 43 / What word would you use when writing the compound inequality for this equation? 6 x - 11 > 18 A B or and

44 Slide 44 / 70 You solve inequalities in the same manner as equations. Use inverse operations until the absolute value bars are alone on one side of the inequality. Next, write the compound inequality. Then, solve each inequality and graph your solution on a number line.

45 Slide 45 / 70 Example 2x - 1 < 5 2x - 1 < 5 and 2x - 1 > -5 Rewrite as a compound inequality 2x < 6 and 2x > -4 Add 1 to each side of the inequality x < 3 and x > -2 Divide each side by 2 Graph the solution

46 Slide 46 / 70 Example 2x > 2 2x + 3 > 9 Add 7 to both sides of the inequality 2x + 3 > 9 or 2x + 3 < -9 Rewrite as a compound inequality 2x > 6 or 2x < -12 Subtract 3 from each side x > 3 or x < -6 Divide each side by 2 Graph the solution

47 Slide 47 / 70 There are special cases to consider when solving absolute value equations. Consider a negative solution. It is not possible for absolute value to be negative. So, if the expression is < or < a negative, there are no solutions. Likewise, if the expression is > or > a negative, there are infinitely many solutions, or the Real Numbers.

48 Slide 48 / 70 Examples: 3x + 2 < -1 No solution It is not possible for the absolute value of an expression to be negative. x + 7 > -1 Infinitely Many Solutions or Real Numbers All absolute value expressions are positive

49 Slide 49 / 70 Try These 3x - 5 < x + 3 > -8

50 Slide 50 / 70 Try These 7x < x + 3 > 1

51 Slide 51 / Solve a + 2 < 7 A a > 5 and a < -9 B a > 5 or a < -9 C a < 5 and a > -9 D a < 5 or a > -9

52 Slide 52 / Solve s - 3 < 13.5 A s > -3 or s < 6 B s > -3 or s > 6 C s > -3 and s < 6 D s > -3 and s < 6

53 Slide 53 / Solve A m > -2 and m < 2 B m > -2 or m < 2 m - 4 > -6 C D no solution real numbers, infinitely many

54 Slide 54 / Solve -5 x < 23 A x > 6 or x < -6 B x > 6 or x < -4 C No Solution D x > 6 and x < -4

55 Slide 55 / 70 Functions Return to Table of Contents

56 Slide 56 / 70 An absolute value function is a function whose rule contains an absolute value expression. For example, f(x) = x - 1 The graphs of absolute value functions have a unique shape. Can you guess what it is?

57 Slide 57 / 70 Remember, functions have exactly one value of the dependent variable for each value of the independent variable. (There is only one y-value for each x-value)

58 Slide 58 / 70 Recall the graph of y = x from when you studied the families of functions

59 Slide 59 / 70 Absolute value functions of the form y = x - b are v-shaped with the vertex at (b, 0).

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63 Slide 63 / Which graph is the solution of the function? y = x - 7 A B C

64 Slide 64 / Which graph is the solution of the function? y = x - 5 A B C

65 Slide 65 / Which graph is the solution of the function? y = 2 x - 1 A B C

66 Slide 66 / Which graph is the solution of the function? y = x + 2 A B C

67 Slide 67 / Which graph is the solution of the function? y = x + 4 A B C

68 Slide 68 / Which equation is represented by the graph below? A y = x 2 3 B y = (x 3) 2 C y = x 3 D y = x 3 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from accessed 17, June, 2011.

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70 Slide 70 / 70 Graph and label the following equations on the set of axes below: y = y = Explain how decreasing the coefficient of x affects the graph of the equation y = From the New York State Education Department. Office of Assessment Policy, Development and administration. Internet. Available from accessed 17, June, 2011.