Inequalities and Absolute Values. Assignment Guide: EOO = every other odd, 1, 5, 9, 13, EOP = every other pair, 1, 2, 5, 6, 9, 10,

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1 Chapter 4 Inequalities and Absolute Values Assignment Guide: E = ever other odd,, 5, 9, 3, EP = ever other pair,, 2, 5, 6, 9, 0, Lesson 4. Page Es , odd, 40-43, 49-52, odd Lesson 4.2 Page Es. 6-27, 30-39, 44-49, 57-60, Lesson 4.3 Page 88-9 Es. 0-8, 22-24, 26-34, 4-45, 50-56, odd Lesson 4.4 Page Es. 3-5, 9-2, 25-32, 37-43, 48-50, odd Lesson 4.5 Page Es. -3, 7-25, 30-39, 46-52, 6-69 odd Lesson 4.6 Page Es. 0-4, 9-23, , 45-48, odd Chapter Review Page Es. -34 Test is on and covers sections

2 4. Solving Linear Inequalities Goal p Solve and graph simple and compound inequalities in one variable. Your Notes Vocabular Linear inequalit in one variable An inequalit that can be written in one of the following forms, where a and b are real numbers and a Þ 0: a b, 0, a b # 0, a b. 0, a b $ 0 of an inequalit in one variable A value of the variable that makes the inequalit true Graph of an inequalit in one variable All points on a real number line that are solutions of the inequalit Compound inequalit Two simple inequalities joined b the word and or the word or Properties of Inequalities To write an equivalent inequalit: Add the same number to each side. Subtract the same number from each side. Multipl or divide each side b the same positive number. Multipl or divide each side b the same negative number and reverse the inequalit smbol. 68 Lesson 4. Algebra 2 C&S Notetaking Guide Copright McDougal Littell/Houghton Mifflin Compan

3 Eample Inequalit with a Variable on ne Side Solve the inequalit. a. 2 6 b. 23z 2 7 < 8 a. 2 6 Write original inequalit. 4 Subtract 2 from each side. The solution is all real numbers less than or equal to 4. b. 23z 2 7 < 8 Write original inequalit. 23z < 5 z > 25 Add 7 to each side. Divide each side b 23 ; reverse the inequalit. The solution is all real numbers greater than 25. Eample 2 Solve 4 5 > Graph the solution. 4 5 > > > 25 Inequalit with a Variable on Both Sides Write original inequalit. Subtract 9 from each side. Subtract 5 from each side. < 3 Divide each side b 25 ; reverse the inequalit. The solution is all real numbers less than 3. The graph is shown below Copright McDougal Littell/Houghton Mifflin Compan Lesson 4. Algebra 2 C&S Notetaking Guide 69

4 Checkpoint Solve the inequalit. Then graph our solution > < Eample 3 Solve an and Compound Inequalit Solve 27 < Then graph the solution. 27 < Write original inequalit < Add 2 to each epression. 25 < 5 0 Simplif. 2 < 2 Divide each epression b 5. The solution is all real numbers greater than 2 and less than or equal to Lesson 4. Algebra 2 C&S Notetaking Guide Copright McDougal Littell/Houghton Mifflin Compan

5 Eample 4 Solve an or Compound Inequalit Solve or Then graph the solution. Solve each part separatel. First Inequalit Second Inequalit Write Write inequalit. inequalit. 4 2 Add 7 to 3 2 Subtract 2 each side. from each side. 3 Divide each 7 Divide each side b 4. side b 3. The solution is all real numbers less than or equal to 3 or greater than or equal to Checkpoint Solve the inequalit. Then graph our solution < 4 5 < < < 4 Homework or 5 > or > 5 Copright McDougal Littell/Houghton Mifflin Compan Lesson 4. Algebra 2 C&S Notetaking Guide 7

6 4.2 Linear Inequalities in Two Variables Goal p Solve and graph linear inequalities in two variables. Your Notes Vocabular Linear inequalit in two variables An inequalit that can be written in one of these forms, where A, B, and C are constants: A B < C, A B C, A B > C, A B C Half-plane Either of the two regions into which the boundar line of a linear inequalit divides the coordinate plane Eample Check s of Inequalities Check whether the given ordered pair is a solution of 4 2 > 6. a. (3, 2) b. (2, 4) RDERED PAIR SUBSTITUTE CNCLUSIN a. (3, 2) 4( 3 ) 2( 2 ) 5 6 > 6 (3, 2) is a solution. b. (2, 4) 4(2) 2( 4 ) 5 4 > 6 (2, 4) is not a solution. Checkpoint Check whether the given ordered pair is a solution of (6, 2) 2. (3, 2) not a solution solution 72 Lesson 4.2 Algebra 2 C&S Notetaking Guide Copright McDougal Littell/Houghton Mifflin Compan

7 Graphing a Linear Inequalit Step Graph the boundar line of the inequalit. Use a dashed line for < or >. Use a solid line for or. Step 2 Test a point that is not on the boundar line to see whether it is a solution of the inequalit. Then shade the appropriate half-plane. Eample 2 Graph Linear Inequalities in ne Variable Graph 2 in a coordinate plane.. Graph the boundar line 5 2. Use a solid line because 2. $ 2 2. Test the point (0, 0). It is a solution, so shade the half-plane that does contain (0, 0). Eample 3 Graph Linear Inequalities in Two Variables Graph < 26 in a coordinate plane.. Graph the boundar line Use a dashed line because < Test the point (0, 0). Because (0, 0) is not a solution, shade the half-plane that does not contain (0, 0) , 26 Copright McDougal Littell/Houghton Mifflin Compan Lesson 4.2 Algebra 2 C&S Notetaking Guide 73

8 Checkpoint Graph the inequalit in a coordinate plane. 3. < 22 4., 22 $ > # < 28 Homework 5 2 $ Lesson 4.2 Algebra 2 C&S Notetaking Guide Copright McDougal Littell/Houghton Mifflin Compan

9 4.3 Sstems of Linear Inequalities Goal p Graph, write, and use a sstem of linear inequalities. Your Notes Vocabular Sstem of linear inequalities in two variables Two or more inequalities in the same variables of a sstem of linear inequalities An ordered pair (, ) that is a solution of each inequalit in the sstem Graph of a sstem of linear inequalities The graph of all solutions of the sstem Eample Check s of Inequalities Check whether (3, 22) is a solution of the sstem of inequalities. a. < 5 b < 29 RDERED PAIR SUBSTITUTE CNCLUSIN a. (3, 22) 3 ( 22 ) 5 < 5 (3, 22) is a 2( 3 ) solution. b. (3, 22) 4( 3 ) ( 22 ) 5 0 (3, 22) is not ( 22 ) 5 2 < 29 a solution. Checkpoint Check whether (25, 4) is a solution of the sstem of inequalities.. 2 > > 0 2 < solution not a solution Copright McDougal Littell/Houghton Mifflin Compan Lesson 4.3 Algebra 2 C&S Notetaking Guide 75

10 Graphing a Sstem of Linear Inequalities Step Graph the boundar lines of the inequalities. Use a dashed line for an inequalit with < or >. Use a solid line for an inequalit with or. Step 2 Shade the half-planes for the inequalities. The graph of the sstem is the region common to all the half planes., 2 The double-shaded region shown above is the graph of < 2 and. $ Eample 2 Graph the sstem. Graph a Sstem of Two Inequalities < 2 Inequalit Inequalit 2. Graph the boundar line of each inequalit. Use a dashed line for Inequalit. Use a solid line for Inequalit Shade the half-plane below 5 2 with horizontal lines to represent Inequalit. # Shade the half-plane on and below with vertical lines to represent Inequalit 2. The graph of the sstem is the overlap, or intersection, of the two shaded areas., 2 76 Lesson 4.3 Algebra 2 C&S Notetaking Guide Copright McDougal Littell/Houghton Mifflin Compan

11 Beginning here in Eample 3, it is onl necessar to show the solution region in our graphs of the sstems of inequalities. Eample 3 Graph the sstem. Graph a Sstem of Three Inequalities > 22 Inequalit 3 Inequalit > 6 Inequalit 3 The inequalit > 22 implies that the region is to the right of the dashed line The inequalit 3 implies that the region is on and below the line 5 3. The inequalit 23 2 > 6 implies that the region is above the dashed line The graph of the sstem is the shaded triangular region shown above. Checkpoint Graph the sstem. 3. < > 2 2 Homework Copright McDougal Littell/Houghton Mifflin Compan Lesson 4.3 Algebra 2 C&S Notetaking Guide 77

12 4.4 Solving Absolute Value Equations Your Notes Goal p Solve and write absolute value equations in one variable. Vocabular Absolute value The distance a number is from 0 on a number line (written ) Absolute value equation An equation that contains an absolute value epression Solving an Absolute Value Equation The absolute value equation a b 5 c where c > 0 is equivalent to the compound statement a b 5 c or a b 5 2c. EXAMPLE EQUIVALENT FRM SLUTINS or , 23 So, the solutions of are 3 and 23. Eample Solve Solve an Absolute Value Equation Rewrite the absolute value equation in two linear equations. Then solve each equation Write original equation or Epression can equal 6 or or 5 23 Add to 3 each side. The equation has two solutions: 9 and 23. CHECK ? ? 6 9 5? ? Lesson 4.4 Algebra 2 C&S Notetaking Guide Copright McDougal Littell/Houghton Mifflin Compan

13 Eample 2 Solve an Absolute Value Equation Solve First isolate the absolute value epression on one side of the equation Write original equation. Subtract 2 from each side. Both solutions should be checked in the original absolute value equation. Rewrite as two linear equations. Then solve each equation or Write original equation. Epression can equal 2 or or Subtract 4 from each side. 5 4 or 5 28 Divide each side b 2. The equation has two solutions: 4 and 28. Checkpoint Solve the equation and and 5 Copright McDougal Littell/Houghton Mifflin Compan Lesson 4.4 Algebra 2 C&S Notetaking Guide 79

14 Eample 3 Write an absolute value equation that has 22 and 8 as its solutions. Write an Absolute Value Equation Graph the numbers on a number line. Then locate the midpoint of the graph. 5 units 5 units The graph of each solution is 5 units from the midpoint, 3. The distance between a number and 3 on a number line is 2 3. You can use the midpoint and the distance to write an absolute value equation. Midpoint Distance An equation that has 22 and 8 as its solutions is CHECK Checkpoint Write an absolute value equation that has the given solutions and and 6 Homework Lesson 4.4 Algebra 2 C&S Notetaking Guide Copright McDougal Littell/Houghton Mifflin Compan

15 4.5 Solving Absolute Value Inequalities Goal p Solve and graph absolute value inequalities. Your Notes Vocabular Absolute value inequalit An inequalit that contains an absolute value epression Solving Absolute Value Inequalities In the inequalities below, c > 0. INEQUALITY EQUIVALENT FRM GRAPH a b < c 2c < a b < c a b c 2c a b c a b > c a b < 2c or a b > c a b c a b 2c or a b c Eample Solve an Inequalit of the Form b c Solve Then graph the solution Write original inequalit Write equivalent compound inequalit Add.5 to each epression. The solution is all real numbers greater than or equal to 23 and less than or equal to Copright McDougal Littell/Houghton Mifflin Compan Lesson 4.5 Algebra 2 C&S Notetaking Guide 8

16 Eample 2 Solve an Inequalit of the Form a b < c Solve 2 8 < 2. Then graph the solution. 2 8 < 2 Write original inequalit. 22 < 2 8 < 2 Write equivalent compound inequalit. 20 < 2 < 26 Subtract 8 from each epression. 25 < < 23 Divide each epression b 2. The solution is all real numbers greater than 25 and less than Checkpoint Solve the inequalit. Then graph our solution < 9 22 < < Lesson 4.5 Algebra 2 C&S Notetaking Guide Copright McDougal Littell/Houghton Mifflin Compan

17 Eample 3 Solve an Inequalit of the Form a b c Solve Then graph the solution. The absolute value inequalit is equivalent to or First Inequalit Second Inequalit Write inequalities Subtract 5 from each side Divide each side b 2. 2 The solutions are all real numbers less than or equal to 24 or greater than or equal to Eample 4 Write a Model for Tolerance Manufacturing A manufacturer of sewing machine needles uses a tolerance of 0.0 millimeter for a needle that is designed to be 3.8 millimeters long. Write and solve an absolute value inequalit that describes the acceptable lengths for the needles Write algebraic model Write equivalent compound inequalit Add 3.8 to each side. The acceptable lengths for the needles is between 3.79 millimeters and 3.8 millimeters, inclusive. Homework Checkpoint Complete the following eercise. 3. Solve > 2. Then graph the solution. < 22 or > Copright McDougal Littell/Houghton Mifflin Compan Lesson 4.5 Algebra 2 C&S Notetaking Guide 83

18 4.6 Absolute Value Functions Goal p Evaluate, graph, and use simple absolute value functions. Your Notes Vocabular Verte The common endpoint of the two ras in the graph of an absolute value function Eample Evaluate the function when 5 24 and 5 7. a. f() 5 3 b. g() a. When 5 24: f() 5 3 Write function. f(24) Substitute 24 for. 5 2 Simplif. 5 When 5 7: f() 5 3 Write function. f(7) Substitute 7 for. 5 0 b. When 5 24: Evaluate Absolute Value Functions 5 0 Simplif. g() g(24) When 5 7: g() g(7) Lesson 4.6 Algebra 2 C&S Notetaking Guide Copright McDougal Littell/Houghton Mifflin Compan

19 Checkpoint Evaluate the function when 5 22 and when f() g() 5 0 9; 2 2; 5 Eample 2 Graph an Absolute Value Function Graph 5 } 3.. Plot the verte at the origin. 2. Find a second point b substituting an -value. 5 } Write original equation. 3 5 } 3 Substitute 3 for. 3 5 } ( 3 ) 3 Evaluate 3. 5 Multipl. A second point is ( 3, ). 3. Plot the second point and use smmetr to plot a third point at ( 23, ). 4. Connect these three points with a V -shaped graph. Copright McDougal Littell/Houghton Mifflin Compan Lesson 4.6 Algebra 2 C&S Notetaking Guide 85

20 Eample 3 Graph Graph an Absolute Value Function. Plot the verte at the origin. 2. Find a second point b substituting an -value Write original equation Substitute for. 5 23( ) Evaluate Multipl. A second point is (, 23 ). 3. Plot the second point and use smmetr to plot a third point at ( 2, 23 ). 4. Connect these three points with a V-shaped graph. Checkpoint Graph the function } 4 Homework 86 Lesson 4.6 Algebra 2 C&S Notetaking Guide Copright McDougal Littell/Houghton Mifflin Compan

21 Words to Review Use our own words and/or an eample to eplain the vocabular word. Linear inequalit in one variable 2 3 $ 5 Graph of an inequalit in one variable 2 Linear inequalit in two variables Sstem of linear inequalities Two or more inequalities in the same variables Graph of a sstem of linear inequalities The graph of all solutions of the sstem Absolute value equation An equation that contains an absolute value epression 3 of an inequalit in one variable $ Compound inequalit Two simple inequalities joined b the word and or the word or Half-plane Either of the two regions into which a boundar line divides the plane of a sstem of linear inequalities An ordered pair that is a solution of each inequalit Absolute value of The distance a number is from 0 on a number line (written ) Absolute value inequalit An inequalit that contains an absolute value epression Verte The common endpoint of the two ras in the graph of an absolute value function Review our notes and Chapter 4 b using the Chapter Review on pages of our tetbook. Copright McDougal Littell/Houghton Mifflin Compan Words to Review Algebra 2 C&S Notetaking Guide 87