"Properties of Equality and Properties of Inequalities" Quadratic functions, equations and inequalities of one variable PPT

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"Properties of Equality and Properties of Inequalities" Quadratic functions, equations and inequalities of one variable PPT

Part One: Explanation of Curriculum Standards

1. Be able to use inequality groups to express inequality relationships.

2. Can be used as a difference method to compare the size of two numbers or expressions.

3. Master the properties of equations.

4. Understand the concept of inequalities and master the properties of inequalities.

5. Be able to use the properties of inequalities to prove inequalities or solve range problems.

PPT on the properties of equations and inequalities, part 2: independent preview

1. Inequality and inequality relationship

1. Fill in the blanks

Inequality and inequality relations

(1) There are two key points in the definition of inequality.

①Unequal symbols >,<,≥,≤ or ≠.

②The relationship expressed is an unequal relationship.

(2) Conversion between literal language and symbolic language in inequalities.

2. Do it

The speed limit on a certain road section is 40 km/h, which means that when the driver is driving on this road section, the speed v (unit: km/h) of the car should not exceed 40 km/h. The inequality written is _________.

Answer:v≤40

2. Comparison of the sizes of real numbers

1. If real numbers a and b are given, how to compare their sizes?

Tip: Their sizes are usually compared by judging the sign of their difference (a-b). When a and b have the same sign and neither is 0, their sizes can also be compared by the relationship between their quotient and 1.

2. Fill in the blanks

The basis for comparing the sizes of real numbers a and b

3. Do it

If x is a real number, then the relationship between x2-1 and 2x-5 is___________.

Analysis: ∵(x2-1)-(2x-5)=x2-2x+4=(x-1)2+3>0,∴x2-1>2x-5.

Answer:x2-1>2x-5

3. Important Inequalities

1. What is the relationship between ∀a, b∈R, a2+b2 and the size of 2ab?

Tip: Because a2+b2-2ab=(a-b)2≥0 is always true, so a2+b2≥2ab.

2. Fill in the blanks

∀a,b∈R,a2+b2≥2ab, if and only if a=b, the equal sign holds.

4. Properties of Inequalities

1. Please sort out the basic properties of the equation and write down the relationship expressions of its symmetry, transitivity, addition, subtraction, multiplication and division.

Tips: (1) Symmetry: if a=b, then b=a;

(2) Transitivity: if a=b, b=c, then a=c;

(3) Additivity: if a=b, then a±c=b±c;

(4) Multiplicability: if a=b, then ac=bc;

PPT on the properties of equations and inequalities, part three: inquiry learning

Express inequality relations using inequalities (groups)

Example 1 It is known that the vitamin A and B contents of two foods A and B are as follows:

Suppose x kg of food A and y kg of food B are used to form a mixed food, and the mixed food contains at least 56 000 units of vitamin A and 63 000 units of vitamin B. Use the inequality set to express the satisfaction of x and y unequal relationship.

Analysis: List the inequalities based on the fact that vitamins A and B are at least 56,000 units and 63,000 units respectively.

Solution: From the meaning of the question, we know that x kg of type A food contains 600x units of vitamin A and 800x units of vitamin B. y kg of type B food contains 700y units of vitamin A and 400y units of vitamin B. Then x kg of type A food The mixed food composed of food and y kg of food B contains a total of vitamin A (600x+700y) units and vitamin B (800x+400y) units.

Reflection and Insight 1. The inequality relationship emphasizes the relationship between quantities, which can be expressed by the symbols ">", "<", "≠", "≥" or "≤"; inequality is used to express the inequality relationship The formula can be expressed by the equations "a>b", "a

2. Steps to use inequalities (groups) to express inequality relationships:

(1) Review the meaning of the question and clarify the number of unequal relationships in the conditions;

(2) Appropriately set unknown numbers to represent variables;

(3) Express each inequality relationship with an inequality and write it in the form of an inequality group.

Variant training 1 When a natural gas company in a certain city installed natural gas pipelines in some residential areas, it adopted a charging method to encourage residents to use natural gas. If every household in the entire community is installed, an overall initial installation fee of 10,000 yuan will be charged, and then each household will be charged 500 yuan. After all households in a community install natural gas according to this charging method, the average payment per household is less than 1,000 yuan. Then the number of households in this community is ()

A. At least 20 households B. At most 20 households

C. At least 21 households D. At most 21 households

Analysis: Suppose the number of households in this community is .

Answer:C

Comparison of real number sizes

Example 2 Compares the magnitude of two algebraic expressions in each of the following groups:

(1)2x2+3 and x+2,x∈R;

Analysis: Use the difference method for comparison. When solving question (2), pay attention to the classification and discussion of the real number a.

PPT on the properties of equations and inequalities, part 4: thinking analysis

Ignoring the condition for taking the equal sign when applying the inequality property leads to an error

Typical example: Suppose f(x)=ax2+bx, and 1≤a-b≤2, 2≤a+b≤4, find the value range of 4a-2b.

Correct answer: Method 1 (Undetermined coefficient method)

Let 4a-2b=m(a-b)+n(a+b),

Then 4a-2b=(m+n)a+(-m+n)b,

So 4a-2b=3(a-b)+(a+b).

Because 1≤a-b≤2, so 3≤3(a-b)≤6.

And 2≤a+b≤4,

So 5≤3(a-b)+(a+b)≤10.

That is, 5≤4a-2b≤10.

Misunderstanding Warning: Finding the value range of a number (or expression) is an important part of the application of the properties of inequalities. When solving problems, the conditional expression should be regarded as a whole and used to express the amount of the range sought. At the same time, pay attention to the conditions for taking the equal sign. Do you have it? You must not use the properties of inequalities to find the range of the variables themselves, and then find the value range of the algebraic expression formed by it. This will often expand the range of the algebraic expression.

PPT on the properties of equations and inequalities, Part 5: Practice in class

1. Which of the following statements is correct ()

A. Someone’s monthly income x is no more than 2,000 yuan, which can be expressed as “x<2,000”

B. If Xiao Ming’s height is x and Xiao Hua’s height is y, then Xiao Ming is shorter than Xiao Hua, which means “x>y”

C. A certain variable x is at least a and can be expressed as "x≥a"

D. A certain variable y does not exceed a and can be expressed as "y≥a"

Answer:C

2. If the real numbers a and b satisfy the condition a>b, then the following inequality must be true ()

Analysis: For A, when a=1, b=-1, it is true, so A is wrong; for B, when a=1, b=-2, a2b, a3>b3 must hold, then D is correct.

Answer:D

3.(x+5)(x+7)________(x+6)2.(Fill in ">""<""≥" or "≤")

Analysis: (x+5)(x+7)-(x+6)2=x2+12x+35-(x2+12x+36)=-1<0, so (x+5)(x+7) <(x+6)2.

Answer:<

4. It is known that 1≤a≤2, 3≤b≤6, then the value range of 3a-2b is ________.

Analysis: ∵1≤a≤2, 3≤b≤6, ∴3≤3a≤6, -12≤-2b≤-6. From the properties of the inequality, we get -9≤3a-2b≤0, that is, 3a-2b The value range is [-9,0].

Answer:[-9,0]

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Update Time: 2024-10-02

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