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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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Chapter 28

INEQUALITIES

不等式

FYHS-Kulai by Chtan


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.

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Some properties of inequalities

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2

3

4

5

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6

7

9.

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10.

11.

12.

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13.

have same sign.

have opposite sign.

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(8)

To prove

Proof :

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Proof :

(9)

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(10)

Proof :

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Equality holds when

2

Equality holds when

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3

Equality holds when

4

Equality holds when

AM-GM inequality

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5

Equality holds when

6

Equality holds when

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7

Equality holds when

Equality holds when

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e.g.1

Consider the function

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Soln :

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9

10

1

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13

14

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Exponential inequalities

To solve this inequality, it is equivalent to solve :

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logarithmic inequalities

It is equivalent to solve :

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It is equivalent to solve :

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e.g.2

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e.g.3

:

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e.g.4

Find the range of values of x for which :

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e.g.5

Find the range of values of x for which :

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e.g.6

:

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e.g.7

Express in the modulus form :

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e.g.8

Express in the modulus form :

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e.g.9

For what values of x is :

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e.g.10

:

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e.g.11

:

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e.g.12

:

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e.g.13

:

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e.g.14

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e.g.15

.

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e.g.16

For what values of x is :

positive .

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e.g.17

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

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e.g.18

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

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e.g.19

For what values of x is :

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e.g.20

Solve the inequality :

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e.g.21

For what values of x is :

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Harder examples

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e.g.22

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Soln :

Using AM-GM inequality, consider the 3 numbers :

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Similarly,

and

Adding the 3 inequalities,

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e.g.23

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Soln :

(i) Given

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

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(ii)

=2

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Since

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2+9+0-27 -3

-9

-6

27

2

3

-9

0

-3

-6

+9

-3

2

0

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Q:

.

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The notorious cases :

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When will you use this ?

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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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Addendum

附录

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1. reduction to absurdity

-- proof by contradiction

反证法

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e.g.

Property (12),

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Soln :

If

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If

Proved.

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2. Solve the inequalities :

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3. Solve the inequalities :

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4. Solve the inequalities :

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5. Solve the inequalities :

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6. Solve the irrational inequalities :

e.g.

e.g.

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The end

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