"Inequality" Equality and Inequality PPT (Lesson 2 Inequalities and Their Properties)

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"Inequality" Equality and Inequality PPT (Lesson 2 Inequalities and Their Properties)

Part One: Learning Objectives

1. Master the properties of inequalities. (emphasis)

2. Can use the properties of inequalities to compare numbers or expressions or prove inequalities. (difficulty)

3． Explore the commonalities and differences between equations and inequalities by analogy.

core competencies

1. Cultivate logical reasoning ability through judgment and proof of the properties of inequalities.

2. Use the properties of inequalities to solve range problems and improve your mathematical literacy.

Inequality PPT, part 2 content: independent preview to explore new knowledge

A preliminary exploration of new knowledge

Basic properties of inequalities

(1) Symmetry: a>b⇔_____.

(2) Transitivity: a>b, b>c⇒_____.

(3) Additivity: a>b⇔_____.

(4) Multiplicability: a>b, c>0⇒_____; a>b, c<0⇒_____.

(5) Addition rule: a>b, c>d⇒_____.

(6) Multiplication rule: a>b>0, c>d>0⇒ _____.

(7) Multiplication rule: a>b>0⇒__________.

First try

1. If a>b, c>d, then one of the following inequality relations is not necessarily true ()

A． a－b＞d－c　B． a＋d＞b+c

C. a-c>b-c D． a-c＜a-d

2. The inequality equivalent to a>b is ()

A． |a|>|b| B. a2>b2

C.ab>1 D. a3>b3

3． Assuming x

A． x2a2

C. x2ax

Inequality PPT, the third part: cooperative exploration to improve literacy

Use the properties of inequalities to determine whether propositions are true or false

[Example 1] For the real numbers a, b, c, the true proposition among the following propositions is ()

A． If a>b, then ac2>bc2

B． If a>b>0, then 1a>1b

C. If aab

D. If a>b, 1a>1b, then a>0, b<0

[Ideas]In this question, you can use the properties of inequalities to directly determine whether the proposition is true or false, or you can use the special value method to determine.

regular method

When using the properties of inequalities to judge, you must pay attention to the conditions for the establishment of the inequality, and do not weaken the conditions. In particular, you cannot fabricate properties at will based on your assumptions. When solving multiple-choice questions about inequalities, you can also use the special value method to eliminate them. Note that the values ​​must be as follows: Principles: First, satisfy the problem setting conditions; second, the value should be simple to facilitate verification and calculation.

Use inequality properties to prove simple inequalities

【Example 2】If a＞b＞0, c＜d＜0, e＜0, prove: ea－c2＞eb－d2.

[Ideas Enlightenment] You can combine the basic properties of inequalities, analyze the structure of the proved inequalities, and derive the proof results with reasonable evidence.

regular method

Things to note when using the properties of inequalities to prove inequalities

1 Some inequalities can be proved by using the properties of inequalities and their corollaries. To solve such problems, one must first understand and memorize the properties of inequalities, and pay attention to applying them flexibly and accurately in solving problems.

2. When applying the properties of inequalities for derivation, attention should be paid to closely following the conditions for the establishment of the properties of inequalities. You must not omit conditions or skip derivation, let alone construct properties and laws at will.

Application of Inequality Properties

[Inquiry Questions]

1. Xiao Ming made the following changes when solving the problem:

∵2

∴13<1b<12,

Also ∵-6

∴－2

Do you think it is correct? Why?

2. From -6

Tip: Incorrect. Because inequalities in the same direction are additive. But you cannot subtract. When solving problems, you must make full use of the conditions and use the properties of inequalities to perform equivalent transformations, and do not "create" properties at will.

Class summary

1. When applying the properties of inequalities, you must understand the prerequisites for their establishment, and do not strengthen or weaken the conditions for their establishment.

2. Pay attention to whether the "arrow" is one-way or two-way, that is, whether each property is reversible.

Inequality PPT, part 4 content: Achieve standards in class and solidify double basics

1. Thinking and analysis

(1) If a>b, then ac>bc must be true. ()

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

[Prompt](1)Error. From the multiplicability of the inequality, we know that when both sides of the inequality are multiplied by a negative number, the direction of the inequality sign changes. Therefore, if a>b, ac>bc does not necessarily hold true.

(2)Error. Take a=4, c=5, b=6, d=2. It satisfies a+c>b+d, but does not satisfy a>b.

2. If a>b>0, c>d>0, then which of the following inequalities is incorrect ()

A． a－d＞b－c　B. -ad<-bc

C. a＋d＞b+c D. ac>bd

3． If -1＜α＜β＜1, then among the following equations, () is always true

A． -2＜α-β＜0 B． -2＜α-β＜-1

C. -1＜α-β＜0 D. -1＜α-β＜1

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Update Time: 2024-07-01

This template belongs to Mathematics courseware People's Education High School Mathematics Edition B Compulsory Course 1 industry PPT template

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