"Inequality" Equality and Inequality PPT (Lesson 1 Inequalities and Their Properties)

Description

"Inequality" Equality and Inequality PPT (Lesson 1 Inequalities and Their Properties)

Part One: Learning Objectives

Be able to use the difference method to compare the magnitude of two numbers or expressions

Master the properties of inequalities and be able to use them to prove inequalities or solve range problems

Inequality PPT, part 2 content: independent learning

Problem guide

Preview the contents of textbook P58-P63 and think about the following questions:

1. How to compare the magnitude of two real numbers?

2. What are the properties of inequalities?

3． What are the corollaries to the properties of inequalities?

A preliminary exploration of new knowledge

1. Compare the sizes of real numbers a and b

(1) Text description

If a-b is a positive number, then a____b; if a-b is zero, then a____b; if a-b is a negative number, then a____b, and vice versa.

(2) Symbol representation

a-b>0⇔a____b; a-b＝0⇔a____b; a-b<0⇔a____b.

■Instructions from famous teachers

The symbol "⇔" is called the equivalence sign, which is pronounced as "equivalent to". The meaning of "p⇔q" is: p can deduce q, and q can also deduce p, that is, p and q can infer each other.

2. Properties of Inequalities

Property 1: If a>b, then a+c____b+c.

Property 2: If a>b, c>0, then ac____bc.

Property 3: If a>b, c<0, then ac____bc.

Property 4: If a>b, b>c, then a____c. (transitive)

Property 5: a>b�b

Corollary 1: If a+b>c, then a____c-b. (Transfer rule of inequalities)

Corollary 2: If a>b, c>d, then a+c____b+d. (Same-way additivity)

Corollary 3: If a>b>0, c>d>0, then ac____bd.

Corollary 4: If a>b>0, then an____bn(n∈N, n>1).

Corollary 5: If a>b>0, then a____b.

■Instructions from famous teachers

(1) Corollary 1 shows that any term in the inequality can move from one side of the inequality to the other side after changing its sign to the opposite sign.

(2) Corollary 2 shows that if both sides of two inequalities in the same direction are added respectively, the resulting inequality will be in the same direction as the original inequality.

(3) Corollary 3 shows that if both sides of n inequalities in the same direction with positive numbers on both sides are multiplied respectively, the resulting inequality will be in the same direction as the original inequality.

self-test

Judge whether it is true or false (mark “√” if it is correct and “×” if it is wrong)

(1) The real number a is not greater than -2, and the inequality is expressed as a≥-2.()

(2) The meaning of the inequality x≥2 means that x is not less than 2.()

(3) If either a

(4) If a+c>b+d, then a>b, c>d.()

Suppose a, b, c∈R, and a>b, then ()

A． ac>bc　B. 1a<1b

C. a2>b2 D. a3>b3

It is known that a>b, c>d, and neither c nor d is 0, then the following inequality must be true ()

A． ad＞bc B. ac>bd

C. a-c>b-d D． a+c>b+d

If x<1, M=x2+x, N=4x-2, then the relationship between M and N is ________.

Inequality PPT, the third part: interactive teaching and practice

Comparison of the size of numbers (formulas)

(1) Compare the sizes of 3x3 and 3x2-x+1;

(2) Given that a≥1, try to compare the sizes of M=a+1-a and N=a-a-1.

regular method

Four steps to compare sizes using the difference method

(1) Make a difference: Make a difference between the two expressions to be compared.

(2) Deformation: Deform the difference expression through general division, factorization, formula and other means.

(3) Judgment symbol: The deformed result is combined with the question setting conditions to determine the error symbol.

(4) Make a conclusion.

[Note]The above steps can be summarized as "three steps and one conclusion", where the "judgment symbol" is the purpose and "deformation" is the key. Among them, there are many deformation techniques, common ones include factorization method, matching method, rationalization method, etc.

Track training

1. If x∈R, y∈R, then ()

A． x2＋y2>2xy－1　B． x2+y2=2xy-1

C. x2+y2<2xy-1 D. x2＋y2≤2xy－1

2. It is known that x>y>0, try to compare the sizes of x3-2y3 and xy2-2x2y.

3． Compare the sizes of 5x2+y2+z2 and 2xy+4x+2z-2.

Properties of Inequalities

(1) For the real numbers a, b, c, there are the following statements:

①If a>b, then ac

②If ac2>bc2, then a>b;

③If aab>b2;

The correct one is ________(fill in the serial number).

(2) If c>a>b>0, prove: ac-a>bc-b.

regular method

Methods of proving inequalities using their properties

(1) The proof of simple inequalities can be obtained directly from known conditions, using the properties of inequalities, and by deforming the inequalities.

(2) For situations where the expressions on both sides of the inequality are relatively complex, it is not easy to prove the properties of the inequality directly. You can consider making a difference on both sides of the inequality, and then deform it, and determine the sign of each factor (formula) according to the conditions. Use the symbol rule to determine the final symbol and complete the proof.

Inequality PPT, Part 4: Feedback on Compliance

1. It is known that b<2a, 3d

A． 2a-c>b-3d B. 2ac>3bd

C. 2a＋c>b+3d D. 2a＋3d>b+c

2. It is known that a1∈(0,1), a2∈(0,1), note M=a1a2, N=a1+a2-1, then the relationship between M and N is ()

A． MN

C. M=N D. M≥N

3． It is known that a and b are real numbers, and a≠b, a<0, then a________2b-b2a. (Fill in “＞”, “＜” or “=")

4. It is known that a, b∈R, x=a3-b, y=a2b-a, try to compare the sizes of x and y.

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