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# Chapter 28 - PowerPoint PPT Presentation

Chapter 28. INEQUALITIES. 不等式. What will be taught in this chapter?. 1. Some fundamental properties of inequalities. 2. Logarithmic function inequalities. 3 . absolute function inequalities. Use the To determine relationship between coeff. and roots. Some properties of inequalities.

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INEQUALITIES

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1. Some fundamental properties of inequalities.

2. Logarithmic function inequalities.

3. absolute function inequalities.

Use the

To determine relationship between coeff. and roots.

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3

4

5

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7

9.

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11.

12.

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have same sign.

have opposite sign.

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To prove

Proof :

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(9)

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Proof :

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2

Equality holds when

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Equality holds when

4

Equality holds when

AM-GM inequality

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Equality holds when

6

Equality holds when

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Equality holds when

Equality holds when

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Consider the function

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10

1

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14

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To solve this inequality, it is equivalent to solve :

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It is equivalent to solve :

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:

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Find the range of values of x for which :

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Find the range of values of x for which :

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:

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Express in the modulus form :

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Express in the modulus form :

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For what values of x is :

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:

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:

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:

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:

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.

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For what values of x is :

positive .

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Find the range of values of x which satisfy the inequality :

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Find the range of values of x which satisfy the inequality :

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For what values of x is :

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Solve the inequality :

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For what values of x is :

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Using AM-GM inequality, consider the 3 numbers :

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and

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(i) Given

Using AM-GM inequality, consider the 4 numbers : :

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=2

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-9

-6

27

2

3

-9

0

-3

-6

+9

-3

2

0

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.

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1. Find the maximum and minimum values of :

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2. Find the range of the function :

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3. Find the possible range of the values of

so that the equation has equal roots, no real roots or 2 real roots.

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4. Show that, if is real, the expression

can take all real values.

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5. Many other cases …

Let me tell when I got the ideas…

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1. reduction to absurdity

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Property (12),

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If

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Proved.

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3. Solve the inequalities :

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5. Solve the inequalities :

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e.g.

e.g.

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