Topicm30cf3f79329b4ced_1528449000663_0Topic

Square inequality

Levelm30cf3f79329b4ced_1528449084556_0Level

Third

Core curriculumm30cf3f79329b4ced_1528449076687_0Core curriculum

III. Equations and inequalities.

The student:

4) solves equations and square inequalities.

Timingm30cf3f79329b4ced_1528449068082_0Timing

45 minutes

General objectivem30cf3f79329b4ced_1528449523725_0General objective

Interpreting and manipulating information presented in the mathematical and popular science texts, as well as in the form of graphs, diagrams, tables.

Specific objectivesm30cf3f79329b4ced_1528449552113_0Specific objectives

1. Communicating in English, developing mathematics skills, scientific, technical and IT competences; developing learning skills.

2. Solving square inequalities.

3. Determining the set of solutions for square inequalities.

Learning outcomesm30cf3f79329b4ced_1528450430307_0Learning outcomes

The student:

1. Student solves square inequalities.

2. Student determines a set of solutions for square inequality.

Methodsm30cf3f79329b4ced_1528449534267_0Methods

1. Problem discussion.

2. Case study.

Forms of workm30cf3f79329b4ced_1528449514617_0Forms of work

1. Individual work.

2. Group work.

Lesson stages

Introductionm30cf3f79329b4ced_1528450127855_0Introduction

Students working in groups organize their knowledge of solving inequalities with one unknown. They give examples of equivalent inequalities, inequalities that do not have a solution and inequalities in which the set of solutions is a closed interval, etc.

Procedurem30cf3f79329b4ced_1528446435040_0Procedure

The teacher informs students that the aim of the lesson will be to solve square inequalities.

Students formulate the definition of square inequality, based on the definition of linear inequality known to them.

Definition
The square inequality (with the unknown $x$) is any inequality, which can be reduced to the form $a{x}^2+bx+c>0$ or $a{x}^2+bx+c\ge 0$ or $a{x}^2+bx+c<0$ or $a{x}^2+bx+c\le 0$ or $a{x}^2+bx+c\le 0$, where $a,b,c$ are fixed real numbers and $a\ne 0$.

Students working in groups, analyze the applet presenting a graphical way to solve square inequalities. They formulate the conclusions.

Task
Analyze the material contained in the applet carefully. What do you notice? Formulate the appropriate conclusions.m30cf3f79329b4ced_1527752263647_0Analyze the material contained in the applet carefully. What do you notice? Formulate the appropriate conclusions.

[Geogebra applet]

The conclusion students should draw.

To solve a square inequality you must:

- find zeros of the appropriate square function (if any),
- sketch the function graph,
- read from the graph for which arguments the function takes positive (negative) values.

Students use new skills to solve tasks.

Task
Solve the inequality graphically - sketch a graph of the appropriate function and read the solution.

1. $-x^2+x+2\le 0$

2. $2x^2-2x-4\ge 0$

3. $3x^2-3x-6<0$

Discussion 
- How can you solve a square inequality written in the form of a product? 
- Which product property should you use?

Students make hypotheses. They formulate their conclusions.

Task
Solve the inequality. What kind of product property will you use? Formulate the conclusion.

1. $\left(x-2\right)\left(2x+6\right)<0$

2. $x(x+5)>0$

3. ${\left(2x-8\right)}^2<0$

Conclusion
To solve the square inequality written in product form, we use the product's properties.
- The product sign depends on the sign of individual factors.
- If both factors are the same characters, then the product is positive.
- If both factors are opposite signs, then the product is negative.m30cf3f79329b4ced_1527752256679_0Conclusion
To solve the square inequality written in product form, we use the product's properties.
- The product sign depends on the sign of individual factors.
- If both factors are the same characters, then the product is positive.
- If both factors are opposite signs, then the product is negative.

Students check their assumptions by analyzing Slideshow presenting the method of solving square inequalities. 

Task
Find the domain of the function $y=-\frac{1}{\sqrt{(3x-x^2)}}$.

Task
There is a square inequality $\left(3x-3\right)\left(2x-a\right)<0$ with the unknown $x$. Calculate the $a$ number, for which the set of solutions for this inequality is the interval (1,4).

Task
A square inequality $\left(3x-3\right)\left(2x-a\right)<0$ is given with an unknown $x$. Determine the $a$ number, for which the only solution of inequality is the number $1\frac{1}{2}$.

Task for volunteers

Calculate the value of the number $m$ so that the domain of the function $y=\sqrt{2x^2-mx+2}$ is the set of real numbers.

Lesson summarym30cf3f79329b4ced_1528450119332_0Lesson summary

Students do the revision exercises. Then they summarize the lesson together, formulating conclusions to remember:

To solve a square inequality:
- calculate zeros of the corresponding square function (if they exist),
- sketch the function graph,
- read from the graph for which arguments the function accepts positive (negative) values.

Selected words and expressions used in the lesson plan

square functionsquare functionsquare function

zeros of a square functionzeros of a square functionzeros of a square function

square inequalitysquare inequalitysquare inequality

a set of solutions of square inequalitiesa set of solutions of square inequalitiesa set of solutions of square inequalities

numerical intervalnumerical intervalnumerical interval

the sum of numerical intervalsthe sum of numerical intervalsthe sum of numerical intervals

an empty setan empty setan empty set

a domain of the functiona domain of the functiona domain of the function