Chapter 1: Tools of Algebra 1-5: Absolute Value Equations and Inequalities Essential Question: What is the procedure used to solve an absolute value equation of inequality? (Tomorrow)

1-5: Absolute Value Equations and Inequalities  A BSOLUTE V ALUE E QUATIONS HAVE TWO SOLUTIONS, because the quantity inside the absolute value sign can be positive or negative  Like compound inequalities, create two equations, and solve them independently. 1. G ET THE ABSOLUTE VALUE PORTION ALONE 2. S ET THE ABSOLUTE VALUE PORTION EQUAL TO BOTH THE POSITIVE AND NEGATIVE

1-5: Absolute Value Equations and Inequalities  Example: Solve |2y – 4| = 12 2y – 4 = 122y – 4 = -12

1-5: Absolute Value Equations and Inequalities  Example: Solve |2y – 4| = 12 2y – 4 = 122y – 4 = y = y = -8

1-5: Absolute Value Equations and Inequalities  Example: Solve |2y – 4| = 12  y = 8 or y = -4  Check:  |2(8) – 4| = |16 – 4| = |12| = 12  |2(-4) – 4| = |-8 – 4| = |-12| = 12 2y – 4 = 122y – 4 = y = 16  2 y = y = -8  2 y = -4

1-5: Absolute Value Equations and Inequalities  Multiple Step Absolute Value Equations  Example 2: Solve 3|4w – 1| – 5 = 10  Get the absolute value portion alone  3|4w – 1| – 5 = 10

1-5: Absolute Value Equations and Inequalities  Multiple Step Absolute Value Equations  Example 2: Solve 3|4w – 1| – 5 = 10  Get the absolute value portion alone  3|4w – 1| – 5 =  3|4w – 1| = 15

1-5: Absolute Value Equations and Inequalities  Multiple Step Absolute Value Equations  Example 2: Solve 3|4w – 1| – 5 = 10  Get the absolute value portion alone  3|4w – 1| – 5 =  3|4w – 1| = 15  3  3  |4w – 1| = 5  Now we can split into two equations, just like the last problem

1-5: Absolute Value Equations and Inequalities  |4w – 1| = 5 4w – 1 = 54w – 1 = -5

1-5: Absolute Value Equations and Inequalities  |4w – 1| = 5 4w – 1 = 54w – 1 = w = w = -4

1-5: Absolute Value Equations and Inequalities  |4w – 1| = 5  w = 1.5 or w = -1  Check (use the original problem):  3|4(1.5) – 1| – 5 = 3|6 – 1| – 5 = 3|5| – 5 = 3(5) – 5 = 15 – 5 = 10  3|4(-1) – 1| – 5 = 3|-4 – 1| – 5 = 3|-5| – 5 = 3(5) – 5 = 15 – 5 = 10 4w – 1 = 54w – 1 = w = 6  4 w = w = -4  4 w = -1

1-5: Absolute Value Equations and Inequalities  Checking for Extraneous Solutions  Sometimes, we’ll get a solution algebraically that fails when we try and check it. These solutions are called extraneous solutions.  Example 3: Solve |2x + 5| = 3x + 4  Is the absolution value portion alone? Yes  When we split this into two equations, we have to NEGATE THE ENTIRE RIGHT SIDE OF THE EQUATION

1-5: Absolute Value Equations and Inequalities  |2x + 5| = 3x + 4 2x + 5 = 3x + 42x + 5 = -3x – 4

1-5: Absolute Value Equations and Inequalities  |2x + 5| = 3x + 4 2x + 5 = 3x + 42x + 5 = -3x – x = 3x – x = -3x – 9

1-5: Absolute Value Equations and Inequalities  |2x + 5| = 3x + 4 2x + 5 = 3x + 42x + 5 = -3x – x = 3x – 1 -3x -x = x = -3x – 9 +3x 5x = -9

1-5: Absolute Value Equations and Inequalities  |2x + 5| = 3x + 4  x = 1 or x = -1.8  You’ll have to check your solutions (next slide) 2x + 5 = 3x + 42x + 5 = -3x – x = 3x – 1 -3x -x = -1  -1 x = x = -3x – 9 +3x 5x = -9  5 x = -1.8

1-5: Absolute Value Equations and Inequalities  |2x + 5| = 3x + 4  x = 1  |2(1) + 5| = 3(1) + 4  |2 + 5| =  |7| = 7 (good)  x = -1.8  |2(-1.8) + 5| = 3(-1.8) + 4  | | =  |1.4| = -1.4 (bad)  The only solution is x = 1  -1.8 is an extraneous solution.

1-5: Absolute Value Equations and Inequalities  Assignment  Page 36  Problems 1 – 15 (all)  You will have to check your solutions for problems 10-15, so show work and identify any extraneous solutions

Chapter 1: Tools of Algebra 1-5: Absolute Value Equations and Inequalities Day 2 Essential Question: What is the procedure used to solve an absolute value equation of inequality?

1-5: Absolute Value Equations and Inequalities  When we solved absolute value equations, we got the absolute value section alone, and set two equations  One as normal  One where we flipped everything outside the absolute value  When solving absolute value inequalities, we do the same thing, except in addition to flipping everything on the other side of the absolute value, FLIP THE INEQUALITY AS WELL  The two lines will always either split apart (greater than) or come together (less than)

1-5: Absolute Value Equations and Inequalities  Example: Solve |3x + 6| > 12. Graph the solution. 3x + 6 > 123x + 6 < -12

1-5: Absolute Value Equations and Inequalities  Example: Solve |3x + 6| > 12. Graph the solution. 3x + 6 > 123x + 6 < x > x < -18

1-5: Absolute Value Equations and Inequalities  Example: Solve |3x + 6| > 12. Graph the solution.  Open circle or closed circle?  Come together or split apart? 3x + 6 > 123x + 6 < x > 6  3 x > x < -18  3 x < -6

1-5: Absolute Value Equations and Inequalities  Example: Solve |3x + 6| > 12. Graph the solution.  Open circle or closed circle? Closed circle (line underneath)  Come together or split apart? Split apart 3x + 6 > 123x + 6 < x > 6  3 x > x < -18  3 x < -6

1-5: Absolute Value Equations and Inequalities  Solve 3|2x + 6| - 9 < 15. Graph the solution.  Need to get the absolute value alone first.  3|2x + 6| - 9 <  3|2x + 6| < 24  3  3  |2x + 6| < 8

1-5: Absolute Value Equations and Inequalities  |2x + 6| < 8 2x + 6 < 82x + 6 > -8

1-5: Absolute Value Equations and Inequalities  |2x + 6| < 8 2x + 6 < 82x + 6 > x < x > -14

1-5: Absolute Value Equations and Inequalities  |2x + 6| < 8  Open circle or closed circle?  Come together or split apart? 2x + 6 < 82x + 6 > x < 2  2 x < x > -14  2 x > -7

1-5: Absolute Value Equations and Inequalities  |2x + 6| < 8  Open circle or closed circle? Open circle (no line)  Come together or split apart? Come together 2x + 6 < 82x + 6 > x < 2  2 x < x > -14  2 x > -7

1-5: Absolute Value Equations and Inequalities  Assignment  Page 36  Problems 16 – 27 (all)  Rest of week, Chapter 1 Test  Wednesday: Preview  Thursday: Review  Friday: Test Day