"Properties of Equality and Properties of Inequality" PPT download of quadratic functions, equations and inequalities of one variable

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"Properties of Equality and Properties of Inequality" PPT download of quadratic functions, equations and inequalities of one variable

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.

PPT on the properties of equations and inequalities, part 2: independent preview to explore new knowledge

A preliminary exploration of new knowledge

1. Properties of equations

(1) Property 1 If a=b, then b=a;

(2) Property 2 If a=b, b=c, then a=c;

(3) Property 3 If a=b, then a±c=b±c;

(4) Property 4 If a=b, then ac=bc;

(5) Property 5 If a=b, c≠0, then ac=bc.

2. 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． x2

B． x2>ax>a2

C. x2

D. x2>a2>ax

PPT on the properties of equations and inequalities, 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

[Imagination] 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.

D [ Method 1: ∵c2≥0, when ∴c=0,

There is ac2=bc2, so A is a false proposition;

From a＞b＞0, we have ab＞0⇒aab＞bab⇒1b＞1a,

Therefore, B is a false proposition;

a＜b＜0⇒-a＞-b＞0⇒-1b＞-1a＞0a＜b＜0⇒-a＞-b＞0⇒ab＞ba,

Therefore, C is a false proposition;

a＞b⇒b－a＜01a＞1b⇒1a－1b＞0⇒b－aab＞0ab＜0.

∵a>b, ∴a>0 and b<0, so D is a true proposition.

Method 2: Special value exclusion method.

Taking c=0, then ac2=bc2, so A is wrong.

Take a=2, b=1, then 1a=12, 1b=1.

Since 1a < 1b, B is wrong. Take a=-2, b=-1,

Then ba=12, ab=2, ba

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.

Track training

1. Which of the following propositions is correct ()

A． If a2>b2, then a>b

B． If 1a>1b, then a

C. If ac>bc, then a>b

D. If a

PPT on the properties of equations and inequalities, part 4: Complying with standards in class and solidifying the bases

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.()

[Tips] (1) Error. From the multiplicability of the inequality, we know that when both sides of the inequality are multiplied by a positive number, the direction of the inequality sign does not change. Therefore, if a>b, ac>bc is not necessarily 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

Keywords: Free download of PPT courseware for compulsory course 1 of Mathematics in High School People's Education A version, PPT download of properties of equations and properties of inequalities, PPT download of quadratic function equations and inequalities of one variable, .PPT format;

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Update Time: 2024-09-05

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

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