Algebra 4'6

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Algebra 4.6

• Solving Quadratic Inequalities

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Homework Review
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Homework Review

• 4. x (x 2 ) 1224
• x2 2x - 1224 0
• ( x 36) ( x 34) 0
• x -36, 34
• So -36 and -34 and 34 and 36

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Homework Review

• 6. x (x - 9 ) 736
• x2 - 9x - 736 0
• ( x 32) ( x 23) 0
• x 32, -23
• So 32 and 23 and -23 and -32

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Homework Review

• 8. (702x) (502x ) 4524
• 3500240x 4x2 4524
• 4x2 240x - 1024 0
• (x 64) ( x - 4 )0
• X -64, 4
• So X has to be 4

x
50
x
70
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Homework Review

• 10. 122 202 x2
• 144 400 x2
• 544 x2
• x /- v544
• So X is ?23.3 ft

x
20
12
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Homework Review
x by x

• 12. (18 2x)(13-2x) 176
• 234 - 62x 4x2 176
• 4x2 - 62x 58 0
• 2x2 - 31x 29 0
• ( x 1) (2x 29) 0
• X 1 0r 14.5
• So X 1 in

13
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Homework Review

• 14. (14 2x)(122x) 288
• 168 52x 4x2 288
• 4x2 52x - 120 0
• x2 13x - 30 0
• (x 15) (x 2) 0
• X -15 0r 2
• So X 2 in

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x
14
x
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Solving Quadratic Inequalities

• Compound Inequality Theorem (lesson 2.8)

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Solving Quadratic Inequalities

• Compound Inequality Theorem (lesson 2.8)
• gt results in a disjunction / or statement (x lt n
  or x gt n)

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Solving Quadratic Inequalities

• Compound Inequality Theorem (lesson 2.8)
• gt results in a disjunction / or statement (x lt n
  or x gt n)
• lt results in a conjunction / and statement (n lt x
  lt x)

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Solving Quadratic Inequalities

• Example
• 15x2 26x 80 gt 0
• (The book tells you to solve this by completing
  the square this will allow you to easily rewrite
  as an absolute value equation but the formula
  will work as well.)

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Solving Quadratic Inequalities

• Example
• 15x2 26x 80 gt 0
• to complete the square
• 15x2 26x gt 80

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Solving Quadratic Inequalities

• Example
• 15x2 26x 80 gt 0
• to complete the square
• 15x2 26x gt 80
• x2 (26/15)x gt (80/15)

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Solving Quadratic Inequalities

• Example
• 15x2 26x 80 gt 0
• to complete the square
• 15x2 26x gt 80
• x2 (26/15)x gt (80/15)
• x2 (26/15)x (676/30) gt (80/15) (676/900)

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Solving Quadratic Inequalities

• Example
• 15x2 26x 80 gt 0
• to complete the square
• 15x2 26x gt 80
• x2 (26/15)x gt (80/15)
• x2 (26/15)x (676/30) gt (80/15) (676/900)
• (x (26/30))2 gt (5476/900)

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Solving Quadratic Inequalities

• Example
• 15x2 26x 80 gt 0
• to complete the square
• 15x2 26x gt 80
• x2 (26/15)x gt (80/15)
• x2 (26/15)x (676/30) gt (80/15) (676/900)
• (x (26/30))2 gt (5476/900)
• x (26/30) gt /- 74/30

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Solving Quadratic Inequalities

• Example
• 15x2 26x 80 gt 0
• to complete the square
• 15x2 26x gt 80
• x2 (26/15)x gt (80/15)
• x2 (26/15)x (676/30) gt (80/15) (676/900)
• (x (26/30))2 gt (5476/900)
• x (26/30) gt /- 74/30
• X gt /- (74/30) (26/30)

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Solving Quadratic Inequalities

• Example
• 15x2 26x 80 gt 0
• to complete the square
• 15x2 26x gt 80
• x2 (26/15)x gt (80/15)
• x2 (26/15)x (676/30) gt (80/15) (676/900)
• (x (26/30))2 gt (5476/900)
• x (26/30) gt /- 74/30
• X gt /- (74/30) (26/30)
• Xgt 48/30 or x gt -100/30

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Solving Quadratic Inequalities

• Example
• 15x2 26x 80 gt 0
• to complete the square
• 15x2 26x gt 80
• x2 (26/15)x gt (80/15)
• x2 (26/15)x (676/30) gt (80/15) (676/900)
• (x (26/30))2 gt (5476/900)
• x (26/30) gt /- 74/30
• X gt /- (74/30) (26/30)
• Xgt 48/30 or x gt -100/30
• X lt -10/3 or x gt 8/5

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Solving Quadratic Inequalities

• Example
• 15x2 26x 80 gt 0
• The final solution looks like this)
• X lt -10/3 or x gt 8/5

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Solving Quadratic Inequalities

• Example
• x2 4x 21 lt 0
• (Solve this and get your two solutions)
• X -7 and 3
• (Since this is a conjunction the final solution
  looks like this)
• -7 lt x lt 3

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Additional Practice

• Solve each inequality
• X2 x 20 gt 0
• X2 10x 16 lt 0
• 5x2 10 gt 27x
• 9x2 31x 12 lt 0

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Additional Practice

• Solve each inequality
• X2 x 20 gt 0 x gt 5 or x lt -4
• X2 10x 16 lt 0 2 lt x lt 8
• 5x2 10 gt 27x x gt 5 or x lt 2/5
• 9x2 31x 12 lt 0 -3 lt x lt -4/9

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Expressing as absolute value inequalities

• You must use the completing the square method
• Look at Example 1

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Expressing as absolute value inequalities

• 15x2 26x 80 gt 0
• to complete the square
• 15x2 26x gt 80
• x2 (26/15)x gt (80/15)
• x2 (26/15)x (676/30) gt (80/15) (676/900)
• (x (26/30))2 gt (5476/900)

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Expressing as absolute value inequalities

• 15x2 26x 80 gt 0
• When we completed the square
• (x 26/30)2 gt 5476/900
• Then rewrite as x26/30 gt 74/30
• Or simplified x13/15 gt 37/15

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Practice Expressing as absolute value inequalities

• Start to solve by completing the square and then
  rewrite as an absolute value inequality
• X2 6x - 40 lt 0

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Practice Expressing as absolute value inequalities

• Solve by completing the square and then rewrite
  as an absolute value inequality
• X2 6x - 40 lt 0
• X2 6x 9 lt 40 9

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Practice Expressing as absolute value inequalities

• Solve by completing the square and then rewrite
  as an absolute value inequality
• X2 6x - 40 lt 0
• X2 6x 9 lt 40 9
• (x 3)2 lt 49

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Practice Expressing as absolute value inequalities

• Solve by completing the square and then rewrite
  as an absolute value inequality
• X2 6x - 40 lt 0
• X2 6x 9 lt 40 9
• (x 3)2 lt 49
• x - 3 lt 7