"Inequality" Equality and Inequality PPT (Solution Set of Inequality in Lesson 3 and Solution of Quadratic Inequality of One Variable in Lesson 4)

"Inequality" Equality and Inequality PPT (Solution Set of Inequality in Lesson 3 and Solution of Quadratic Inequality of One Variable in Lesson 4)

Part One: Learning Objectives

1. Master the solution sets of inequalities and the solution sets of inequality groups.

2. Solve absolute value inequalities. (main difficulty)

3． Master the solution of quadratic inequalities of one variable. (emphasis)

4. Able to solve simple problems based on the relationship between "three quadratics". (difficulty)

core competencies

1. Understand absolute value inequalities through mathematical abstraction.

2. Cultivate mathematical operations literacy through the study of quadratic inequalities of one variable.

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

A preliminary exploration of new knowledge

1. Solution sets of inequalities and solutions of groups of inequalities

Generally speaking, the set of all solutions of an inequality is called the solution set of an inequality. For an inequality group obtained by combining several inequalities, the ____ of the solution sets of these inequalities is called the solution set of the inequality group.

2. absolute value inequality

Generally speaking, inequalities containing ____ are called absolute value inequalities.

3. The distance formula between two points on the number axis and the midpoint coordinate formula

Generally speaking, if the corresponding points of real numbers a and b on the number axis are A and B respectively, that is, A(a) and B(b), then the length of line segment AB is AB=|a-b|, This is the formula for the distance between two points on the number line. The coordinate formula of the midpoint of line segment AB on the number axis is x=a+b2.

4. The concept of quadratic inequality of one variable

Generally, an inequality of the form ax2+bx+c>0 is called a quadratic inequality of one variable, where a, b, c are constants, and a≠0.

5. General form of quadratic inequality of one variable

(1)ax2＋bx＋c＞0(a≠0).

(2)ax2＋bx＋c≥0(a≠0).

(3)ax2＋bx＋c＜0(a≠0).

(4)ax2＋bx＋c≤0(a≠0).

Thinking 2: Is the inequality x2-y2>0 a quadratic inequality of one variable?

Tip: This inequality contains two variables. According to the definition of quadratic inequality of one variable, it can be seen that it is not a quadratic inequality of one variable.

6. Solutions and solution sets of quadratic inequalities of one variable

The value of the unknown number that makes a quadratic inequality of one variable true is called the solution of the quadratic inequality of one variable, and the set of its solutions is called the _____ of the quadratic inequality of one variable.

Thinking 3: Analogy "The solution set of the equation x2 = 1 is {1, -1}, and every element in the solution set can make the equation true." What is the solution set of the inequality x2>1 and its meaning?

First try

1. The solution set of the inequality group 2x+1>0 and 3x-2≤0 is ()

A.x－12≤x≤23　B.x12＜x≤23

C.x－12＜x＜23 D.x－12＜x≤23

2. The solution set of inequality 3x2-2x+1>0 is ()

A.x－1＜x＜13　B.x13＜x＜1

C. ∅D． R

3． The solution set of the inequality |x|-3<0 is ________.

4. The solution set of the inequality -3x2+5x-4>0 is ________.

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

Find the solution set of the system of inequalities

[Example 1] The solution set of the inequality group 12x-1≤0, x+3>0 is ()

A． x>-3B． -3≤x<2

C. -3＜x≤2 D． x≤2

C　12x－1≤0，①x＋3＞0，②

Solve inequality ① to get: x≤2, solve inequality ② to get: x>-3,

∴The solution set of the inequality group is -3

regular method

Solving strategies for solution sets of linear inequalities of one variable

(1) The solution set of a linear inequality group of one variable is the intersection of the solution sets of each inequality;

(2) The formula for finding the solution set of an inequality group: If the equation is the same, get the larger. If the equation is the same, get the smaller. Find the middle of the big or small. If you can’t find the big or small, you can’t find it (no solution).

Solve absolute value inequalities

[Example 2] The solution set of the inequality |5-4x|>9 is ________.

xx<-1 or x>72　[∵|5-4x|>9, ∴5-4x>9 or 5-4x<-9.

∴4x＜-4 or 4x＞14,

∴x＜-1 or x＞72.

∴The solution set of the original inequality is xx＜-1 or x＞72.]

regular method

1. Solution of |x|a-type inequalities

Inequality a＞0 a＝0 a＜0

|x|＜a {x|-a＜x＜a} ∅ ∅

|x|＞a {x|x＞a or x＜-a} {x|x∈R and x≠0} R

2. Solutions to inequalities of type |ax＋b|≤c(c＞0) and |ax+b|≥c(c＞0)

(1)|ax＋b|≤c⇔-c≤ax＋b≤c;

(2)|ax+b|≥c⇔ax+b≥c or ax+b≤-c.

Solution to Quadratic Inequality of One Variable

[Example 3] Solve the following inequalities:

(1)2x2＋7x+3>0;

(2)-4x2+18x-814≥0;

(3)-2x2+3x-2<0.

regular method

General steps for solving quadratic inequalities of one variable without parameters

1 standardization. By deforming the inequality, make the right side of the inequality 0 and make the coefficient of the quadratic term positive.

2 Discriminant. Factor the left side of the inequality. If it is not easy to factor, calculate the discriminant of the corresponding equation.

3. Find the real roots. Find the roots of the corresponding quadratic equation or explain whether the equation has real roots based on the discriminant.

4. Draw a sketch. Draw a sketch of the corresponding quadratic function based on the roots of the quadratic equation of one variable.

5. Write the solution set. Write the solution set of the inequality based on the image.

Class summary

1. The solution set of the inequality (group) should be written in the form of a set. The solution set of the inequality group is the intersection of the solution sets of each inequality.

2. The key to solving absolute value inequalities is to remove the absolute value and use the geometric meaning of absolute value inequalities to solve, which embodies the idea of ​​combining numbers and shapes.

3． Common methods for solving quadratic inequalities of one variable

(1) Image method: From the relationship between quadratic equations of one variable, quadratic inequalities of one variable and quadratic functions, the general steps for solving quadratic inequalities of one variable can be obtained:

①Convert the inequality into standard form: ax2+bx+c>0(a>0) or ax2+bx+c<0(a>0);

② Find the roots of the equation ax2+bx+c=0 (a>0), and draw a simplified diagram of the image of the corresponding function y=ax2+bx+c;

③ Obtain the solution set of the inequality from the image.

(2) Algebraic method: Convert the given inequality into a general expression and solve it with the help of decomposition factors or formulas.

When m0, then {x|x>n or x

If (x-m)(x-n)<0, then {x|m

There is a formula as follows: if it is greater, take both sides, if it is less, take the middle.

4. Inequalities of quadratic form with parameters

When solving quadratic inequalities of one variable containing parameters, it is often necessary to classify and discuss the parameters. In order to achieve "no duplication and no omission" of classification, the discussion needs to consider the following three aspects:

(1) Discussion on the types of inequalities: quadratic term coefficient a>0, a<0, a=0.

(2) Discussion on the roots of equations corresponding to inequalities: two roots (Δ>0), one root (Δ=0), and no roots (Δ<0).

(3) Discussion on the size of the roots of the equations corresponding to the inequalities: x1>x2, x1=x2, x1

5. The opening of the quadratic function and the coordinates of the intersection with the x-axis can be deduced from the solution set of the quadratic inequality of one variable.

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

1. Thinking and analysis

(1)mx2－5x<0 is a quadratic inequality of one variable. ()

(2) If a>0, then the quadratic inequality ax2+1>0 has no solution. ()

(3) If the two roots of the quadratic equation ax2+bx+c=0 are x1 and x2 (x1

(4) If the solution set of |x|>c is R, then c≤0.()

[Prompt](1)Error. When m=0, it is a linear inequality of one variable; when m≠0, it is a quadratic inequality of one variable.

(2)Error. Because a>0, the inequality ax2+1>0 always holds, that is, the solution set of the original inequality is R.

(3)Error. When a>0, the solution set of ax2+bx+c<0 is {x|x1

(4) Obviously c=0 does not hold, which is wrong.

2. Given A (3) and B (-5) on the number axis, the coordinates of the midpoint M of line segment AB are ________.

3． If 1x<2 and |x|>13 are simultaneously true, then the value range of x is ________.

4. Solve the following inequalities:

(1)x(7-x)≥12;

(2)x2>2(x-1)．

Keywords: Free download of PPT courseware for compulsory course No. 1 Mathematics of High School People's Education B version, download of inequalities PPT, download of equality and inequalities PPT, download of solution set of inequalities PPT, download of solution of quadratic inequalities of one variable PPT, .PPT format;

For more information about the PPT courseware "Equations and Inequalities, Inequalities, Inequalities, Solution Sets of Inequalities, Solution Sets of Quadratic Inequalities of One Variable," please click the Equations and Inequalities ppt Inequality, ppt Solution Sets of Inequalities, ppt Solution of Quadratic Inequalities of One Variation ppt tag.

"The Relationship between Functions, Equations, and Inequalities" Function PPT Courseware (Lesson 2):

"The Relationship between Functions, Equations, and Inequalities" Function PPT Courseware (Lesson 2) Part One: Learning Objectives 1. Master the existence theorem of function zeros, and be able to determine the number of function zeros. (Key Points) 2. Understand that the bisection method is a common rule for finding approximate solutions to equations..

"The Relationship between Functions, Equations, and Inequalities" Function PPT Courseware (Lesson 1):

"The Relationship between Functions, Equations, and Inequalities" Function PPT Courseware (Lesson 1) Part One: Learning Objectives 1. Understand the concept of zero points of functions and the relationship between zero points of functions and roots of equations. (Difficulty) 2. Can find the zeros of a function. (Key points) 3..

"The relationship between functions, equations and inequalities" Function PPT (the existence of zero points and how to find their approximate values ​​in Lesson 2):

"The Relationship between Functions, Equations, and Inequalities" Function PPT (Existence of Zero Points and How to Find Approximations in Lesson 2) Part One Content: Learning Objectives: Use the existence theorem of function zero points to determine the existence of zero points of a function in a certain interval and the number of zero points will be determined according to...