Chapter%201:%20Tools%20of%20Algebra%201-5:%20Absolute%20Value%20Equations%20and%20Inequalities

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Chapter 1 Tools of Algebra1-5 Absolute Value
Equations and Inequalities

• Essential Question What is the procedure used to
  solve an absolute value equation of inequality?
  (Tomorrow)

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1-5 Absolute Value Equations and Inequalities

• Absolute Value Equations 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.
• Get the absolute value portion alone
• Set the absolute value portion equal to both the
  positive and negative

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1-5 Absolute Value Equations and Inequalities

• Example Solve 2y 4 12

2y 4 12 2y 4 -12

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1-5 Absolute Value Equations and Inequalities

• Example Solve 2y 4 12

2y 4 12 2y 4 -12
4 4 2y 16 4 4 2y -8
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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 12 2y 4 -12
4 4 2y 16 ?2 ?2 y 8 4 4 2y -8 ?2 ?2 y -4
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1-5 Absolute Value Equations and Inequalities

• Multiple Step Absolute Value Equations
• Example 2 Solve 34w 1 5 10
• Get the absolute value portion alone
• 34w 1 5 10

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1-5 Absolute Value Equations and Inequalities

• Multiple Step Absolute Value Equations
• Example 2 Solve 34w 1 5 10
• Get the absolute value portion alone
• 34w 1 5 10 5 5
• 34w 1 15

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1-5 Absolute Value Equations and Inequalities

• Multiple Step Absolute Value Equations
• Example 2 Solve 34w 1 5 10
• Get the absolute value portion alone
• 34w 1 5 10 5 5
• 34w 1 15?3 ?3
• 4w 1 5
• Now we can split into two equations, just like
  the last problem

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1-5 Absolute Value Equations and Inequalities

• 4w 1 5

4w 1 5 4w 1 -5

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1-5 Absolute Value Equations and Inequalities

• 4w 1 5

4w 1 5 4w 1 -5
1 1 4w 6 1 1 4w -4
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1-5 Absolute Value Equations and Inequalities

• 4w 1 5
• w 1.5 or w -1
• Check (use the original problem)
• 34(1.5) 1 5 36 1 5 35 5
  3(5) 5 15 5 10
• 34(-1) 1 5 3-4 1 5 3-5 5
  3(5) 5 15 5 10

4w 1 5 4w 1 -5
1 1 4w 6 ?4 ?4 w 1.5 1 1 4w -4 ?4 ?4 w -1
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1-5 Absolute Value Equations and Inequalities

• Checking for Extraneous Solutions
• Sometimes, well 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

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1-5 Absolute Value Equations and Inequalities

• 2x 5 3x 4

2x 5 3x 4 2x 5 -3x 4

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1-5 Absolute Value Equations and Inequalities

• 2x 5 3x 4

2x 5 3x 4 2x 5 -3x 4
-5 -5 2x 3x 1 -5 -5 2x -3x 9
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1-5 Absolute Value Equations and Inequalities

• 2x 5 3x 4

2x 5 3x 4 2x 5 -3x 4
-5 -5 2x 3x 1 -3x -3x -x -1 -5 -5 2x -3x 9 3x 3x 5x -9
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1-5 Absolute Value Equations and Inequalities

• 2x 5 3x 4
• x 1 or x -1.8
• Youll have to check your solutions (next slide)

2x 5 3x 4 2x 5 -3x 4
-5 -5 2x 3x 1 -3x -3x -x -1 ?-1 ?-1 x 1 -5 -5 2x -3x 9 3x 3x 5x -9 ?5 ?5 x -1.8
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1-5 Absolute Value Equations and Inequalities

• 2x 5 3x 4
• x 1
• 2(1) 5 3(1) 4
• 2 5 3 4
• 7 7 (good)
• x -1.8
• 2(-1.8) 5 3(-1.8) 4
• -3.6 5 -5.4 4
• 1.4 -1.4 (bad)
• The only solution is x 1
• -1.8 is an extraneous solution.

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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

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Chapter 1 Tools of Algebra1-5 Absolute Value
Equations and InequalitiesDay 2

• Essential Question What is the procedure used to
  solve an absolute value equation of inequality?

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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)

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1-5 Absolute Value Equations and Inequalities

• Example Solve 3x 6 gt 12. Graph the solution.

3x 6 gt 12 3x 6 lt -12

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1-5 Absolute Value Equations and Inequalities

• Example Solve 3x 6 gt 12. Graph the solution.

3x 6 gt 12 3x 6 lt -12
-6 -6 3x gt 6 -6 -6 3x lt -18
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1-5 Absolute Value Equations and Inequalities

• Example Solve 3x 6 gt 12. Graph the solution.
• Open circle or closed circle?
• Come together or split apart?

3x 6 gt 12 3x 6 lt -12
-6 -6 3x gt 6 ?3 ?3 x gt 2 -6 -6 3x lt -18 ?3 ?3 x lt -6
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1-5 Absolute Value Equations and Inequalities

• Example Solve 3x 6 gt 12. Graph the solution.
• Open circle or closed circle? Closed circle (line
  underneath)
• Come together or split apart? Split apart

3x 6 gt 12 3x 6 lt -12
-6 -6 3x gt 6 ?3 ?3 x gt 2 -6 -6 3x lt -18 ?3 ?3 x lt -6
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1-5 Absolute Value Equations and Inequalities

• Solve 32x 6 - 9 lt 15. Graph the solution.
• Need to get the absolute value alone first.
• 32x 6 - 9 lt 15 9 9
• 32x 6 lt 24?3 ?3
• 2x 6 lt 8

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1-5 Absolute Value Equations and Inequalities

• 2x 6 lt 8

2x 6 lt 8 2x 6 gt -8

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1-5 Absolute Value Equations and Inequalities

• 2x 6 lt 8

2x 6 lt 8 2x 6 gt -8
-6 -6 2x lt 2 -6 -6 2x gt -14
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1-5 Absolute Value Equations and Inequalities

• 2x 6 lt 8
• Open circle or closed circle?
• Come together or split apart?

2x 6 lt 8 2x 6 gt -8
-6 -6 2x lt 2 ?2 ?2 x lt 1 -6 -6 2x gt -14 ?2 ?2 x gt -7
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1-5 Absolute Value Equations and Inequalities

• 2x 6 lt 8
• Open circle or closed circle? Open circle (no
  line)
• Come together or split apart? Come together

2x 6 lt 8 2x 6 gt -8
-6 -6 2x lt 2 ?2 ?2 x lt 1 -6 -6 2x gt -14 ?2 ?2 x gt -7
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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