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Chapter 28

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Chapter 28

INEQUALITIES

不等式

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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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FYHS-Kulai by Chtan

(10)

Proof :

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FYHS-Kulai by Chtan

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Some common inequalities formulae

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Equality holds when

2

Equality holds when

FYHS-Kulai by Chtan

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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FYHS-Kulai by Chtan

Exponential inequalities

To solve this inequality, it is equivalent to solve :

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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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FYHS-Kulai by Chtan

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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