§4. LINEAR INEQUALITIES IN TWO VARIABLES
Prepared: Đỗ Thúy Quỳnh
Period:
Teaching day:
Class:
I. LESSON’S OBJECTIVE
At the end of lesson, the student will be able to:
- State definitions of linear inequalities in two variables and their solution regions and
give some illustrated examples.
- Show how to determine the solution region of linear inequalities in two variables.
- Produce a geometric representation of solution set ( Graph the solution region) of
the specifical linear inequalities.
II. PREPARATION
1.Teacher: Lesson plan, Algebra textbook 10.
2.Student: Study before the lesson, textbook 10, notebooks and material.
III. ACTIVITIES FOR TEACHING AND LEARNING
1. Motivation/ Warm-up
a, Objective
- Review draw graph of linear fuction and check some ordered pairs : Which belong to
the graph ?
b, Content:Teacher organizes group activities for students to review draw graph the
linear function and check Which ordered pairs belong to the graph ?
c, Result: Students have the graph of fuction: y= 2x-3 and some ordered pairs belong
to the graph.
d, Method and organization:
T: Divide the class into 3 groups and give each group a sheet A0 which is drawn the
coordinate plane and some tasks. Three groups will finish it in 5 minutes.The group
that finishes the earliest will win.
1,Draw the graph of the function y= 2x-3
2, Which of ordered pairs belong to the graph ? ( if the point belongs to the graph ,
please mark it with red ink in coordinate plane and if not please mark it with black
ink.): (5,3), (2,1), (0,0).
S: Work in group.
2. Study new lesson
- Before the lesson, Teacher introduces for students some new vocabularies and
phrases used in lesson.
Vocabularies and phrases
Meaning
Inequality
bất phương trình
Half-plane
nửa mặt phẳng
Test point
điểm lấy kiểm tra
boundary line
đường thẳng biên
solution region
miền nghiệm
S: Listen and read, write those vocabularies
Activity 1: State definition of linear inequalities in two variable.
a,Objective:
- State definitions of linear inequalities in two variables.
- Give some illustrated examples.
b, Content: From first problem in motivative activity , teacher suggests students to
state definitions of linear inequalities in two variables.
c, Result: Student give some examples of linear inequalities in two variables.
d, Method and organization:
Time
Teacher’s activities
-Recall the definition of linear
equation in two variables and give
some examples.
Student’s activities
- A linear equation has form:
𝑎𝑥 + 𝑏𝑦 = 𝑐
where a,b,c are real numbers.
- Give some examples.
2x-y=3, 5x+8y=12
- Remark: If we replace the sign "=" - 𝑎𝑥 + 𝑏𝑦 < 𝑐, 𝑎𝑥 + 𝑏𝑦 > 𝑐,
in 𝑎𝑥 + 𝑏𝑦 = 0 into the sign " >,<, ≤
, ≥", what form can we get?
- Conclusion: These inequalities
𝑎𝑥 + 𝑏𝑦 < 𝑐, 𝑎𝑥 + 𝑏𝑦 > 𝑐,
𝑎𝑥 + 𝑏𝑦 ≤ 𝑐, 𝑎𝑥 + 𝑏𝑦 ≥ 𝑐 are called
linear inequalities in two variables.
𝑎𝑥 + 𝑏𝑦 ≤ 𝑐, 𝑎𝑥 + 𝑏𝑦 ≥ 𝑐
-So, from the above remark, request
-" An inequality in two variables is of
students to state definition of linear
the form:
inequalities in two variables.
𝑎𝑥 + 𝑏𝑦 < 𝑐,
𝑎𝑥 + 𝑏𝑦 > 𝑐,
𝑎𝑥 + 𝑏𝑦 ≤ 𝑐, 𝑎𝑥 + 𝑏𝑦 ≥ 𝑐
where a,b,c are real numbers, and
𝑎2 + 𝑏 2 ≠ 0, x and y are variables.
-Call some students and request them - Give some examples
to give example for logarithmic
equation:
 Question: "Who can give
me some examples of linear
inequalities in two variables
"
 Another one.
Activity 2: Graphing solution region of inequalities
a,Objective:
- State definition of solution region of linear inequalities in two variables.
- Show how to determine the solution region of linear inequalities in two variables.
- Graph the solution region of the specifical linear inequalities.
- Compare the differences between equations and inequalities.
b, Content:
- From first problem in motivative activity , teacher suggests students to state
definitions of solution region of linear inequalities in two variables and guides
students to discover the rules for geometric representation of solution region.
c, Result: Students represent solution region of linear inequalities in two variables on
the coordinate plane.
d, Method and organization:
Time
Teacher’s activities
Student’s activities
* Definition of solution region of
linear inequalities in two variables
- Draw into notebook the graph
-Come back to first problem in
y=2x-3
motivative activity, (use the graph
on sheet A0) , request draw into
notebook the graph y=2x-3
+ Consider the fuction y=2x-3
<=> 2x-y=3 ( 1)
We can say that (1) is an linear
equation.
Teacher remarks that if we replace
the sign "=" into the sign " ≥ ", we
get a
linear inequalities in two variables
2x-y≥3.(2)
- Request students check: Which of
- (5,3) and (2,1) satisfy (2)
(0,0) doesn't satisfy (2)
ordered pairs below satisfy the
inequality (2) ?
( 5,3) (2,1) (0,0)
- Remark: these points like (5,3) and
(2,1) are solutions of inequality (2).
And the set of points with there
coordinates as solutions is called the
solution region.
 " Can you help me state
- State definition of the solution
region of linear inequalities.
definition of the solution
region of linear inequalities?"
- Linear equations almost always give
- From the definition:
 Can you remark the
us a single solution, but linear
inequalities yield a range of ordered
differences between equations pairs ( a region ).
and inequalities,( specifically
guiding students to recognize
set of solution)?
So, How do we do to graph the
solution region of the linear
inequalities on coordinate plane?
* Graph the solution region of the
linear inequalities
- the graph of 2x-y=3 divides the
- Look at the board ( sheet A0 ).
coordinate plane into 2 parts.
 How many parts does the
graph of 2x-y=3 divide the
-The half- plane on oneside
coordinate plane into ?
containing (5,3) is the solution region
 From the above remark , we
of inequality 2x-y ≥ 3.
have ( 5,3) is a solution of
equation 2x-y=3. Which half
of the plane contains the
1
solution region ?
1
- Conclusion, mark the solution
region on coordinate plane with
blue ink.
- Request students to show how to
determine the solution region of
linear inequalities in two variables
-
ax +by ≤ c ( similar to
other case )
 On the Oxy coordinate plane,
draw line d : ax+by=c.
 Take a point Mo (xo ; yo ) which
isn’t on line d.
 Compute
axo +byo and compare it with c.
 Conclusion
- If a𝑥𝑜 +𝑏𝑦𝑜 < c, then the half- plane
on one side of the boundary line d
and containing 𝑀𝑜 is solution domain
of ax +by ≤ 𝑐
- If a𝑥𝑜 +𝑏𝑦𝑜 >c, then the half- plane
on one side of the boundary line d
without containing 𝑀𝑜 is solution
domain of ax +by ≤ 𝑐
- Consider example
- Practice Exp1
 Exp1:Graph the solution
region of the linear
inequality:x-2y> 4
3. Practice
a, Objective
- Review graph the solution region of the linear inequality
b, Content:Teacher gives exercise for students practice
c, Result: Students graph the solution region of the linear specifical inequalities.
d, Method and organization:
T: Practice through exercise:
Exercise: Graph the solution region of the linear inequality
a,2x-4y+5>0
b,-2x+3y-6≤ 0
c, -x+2+2(y-2)< 2(1-x)
S: Practice individually
4. Apply
- Teacher suggests to the application of solution region representations in systems of
equations in next lesson (which will be used to solve economic problems).