qr code generator vb.net 2010 Linear Inequalities in VS .NET

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CHAPTER 9 Linear Inequalities
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12 Joel wants to invest $10,000 Some will be deposited into an account earning 6% interest and the rest into an account earning 71% interest If he wants at least $650 interest each year, how 4 much can he invest at 71% 4 (a) (b) (c) (d) At least $6000 at 71% 4 More than $6000 at 71% 4 At least $4000 at 71% 4 More than $4000 at 71% 4 b
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13 2 < 4 3x < 2 a 2 < x < 2 3 d 14 3 <
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<x<2
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c 2 < x < 2 3
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>x>2
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2x 1 <7 5 a 2 < x < 4 d 8 < x < 18
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b 10 < x < 20
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c 7 < x < 17
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15 The interval notation for x a 1; 6 b 1; 6
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6 is c 1; 6 d 1; 6
Solutions
1 5 9 13 (a) (b) (c) (b) 2 6 10 14 (c) (a) (d) (d) 3 7 11 15 (b) (a) (a) (c) 4 (c) 8 (c) 12 (c)
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Quadratic Equations
A quadratic equation is one that can be put in the form ax2 bx c 0 where a, b, and c are numbers and a is not zero (b and/or c might be zero) For instance 3x2 7x 4 is a quadratic equation 3x2 7x 4 4 4 2 3x 7x 4 0 In this example a 3, b 7, and c 4 There are two main approaches to solving these equations One approach uses the fact that if the product of two numbers is zero, at least one of the numbers must be zero In other words, wz 0 implies w 0 or z 0 (or both w 0 and z 0) To use this fact on a quadratic equation rst make sure that one side of the equation is zero and factor the other side Set each factor equal to zero then solve for x
Examples
x2 2x 3 0 x2 2x 3 can be factored as x 3 x 1 x2 2x 3 0 becomes x 3 x 1 0
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Quadratic Equations
Now set each factor equal to zero and solve for x x 3 0 3 3 x 3 x 1 0 1 1 x 1
You can check your solutions by substituting them into the original equation x2 2x 3 0 x 3: 3 2 2 3 3 9 6 3 0 p x 1: 12 2 1 3 1 2 3 0 x2 5x 6 0 becomes x 2 x 3 0 x 2 0 2 2 x 2 x2 7x 8 8 8 x2 7x 8 0 becomes x 8 x 1 0 x 8 0 8 8 x 8 x 4 0 4 4 x 4 x 1 0 1 1 x 1 x 4 0 4 4 x 4 x 3 0 3 3 x 3 p
x2 16 0 becomes x 4 x 4 0
3x2 9x 30 0 becomes 3 x2 3x 10 0 which becomes 3 x 5 x 2 0 x 5 0 5 5 x 5 x 2 0 2 2 x 2
The factor 3 was not set equal to zero because 3 0 does not lead to any solution
Practice
Quadratic Equations
1: x2 x 12 0 2: x2 7x 12 0 3: x2 8x 15 4: x2 10x 21 5: 3x2 x 2 0 6: 4x2 8x 5 7: x2 25 0 8: 9x2 16 0 9: x2 100 10: x2 6x 9 0 11: x2 0 12: 5x2 0 1 13: x2 0 9
Solutions
1 x2 x 12 0 x 4 x 3 0 x 4 0 4 4 x 4 2 x2 7x 12 0 x 3 x 4 0 x 3 0 3 3 x 3
x 3 0 3 3 x 3 3 x2 8x 15
x 4 0 4 4 x 4
Quadratic Equations
15 15 x 8x 15 0 x 3 x 5 0
x 3 0 3 3 x 3 4 x2 10x 21 21 21 x 10x 21 0 x 3 x 7 0
x 5 0 5 5 x 5
x 3 0 3 3 x 3 5 3x2 x 2 0 3x 2 x 1 0 3x 2 0 2 2 3x 2 2 x 3 6 4x2 8x 5 5 5 4x 8x 5 0 2x 1 2x 5 0
x 7 0 7 7 x 7
x 1 0 1 1 x 1
2x 1 0 1 1 2x 1 1 x 2
2x 5 0 5 5 2x 5 5 x 2
Quadratic Equations
7 x2 25 0 x 5 x 5 0 x 5 0 5 5 x 5 8 9x2 16 0 3x 4 3x 4 0 3x 4 0 4 4 3x 4 4 x 3 9 x2 100 100 100 x2 100 0 x 10 x 10 0 x 10 0 10 10 x 10 10 x2 6x 9 0 x 3 x 3 0 x 3 0 3 3 x 3 11 x2 0 x x 0 x 0 12 5x2 0 5 x x 0 x 0 x 10 0 10 10 x 10 3x 4 0 4 4 3x 4 4 x 3 x 5 0 5 5 x 5
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