Powerpoint Presentation on Linear Inequality in Two Variable

1. QUARTER 2
WEEK 1
MODULE
Prepared by: Evangeline D. Danao
Grade 8 Mathematics

2. To the Learners:
Before starting the module, I want you to set aside other tasks that will
disturb you while enjoying the lessons. Read the simple instructions below to
successfully enjoy the objectives of this kit. Have fun!
1. Follow carefully all the contents and instructions indicated in every page
of this module.
2. Writing enhances learning. Keep this in mind and take note of the
important concepts in your notebook.
3. Perform all the provided activities in the module.
4. Let your facilitator/guardian assess your answers using the answer key
card.
5. Analyze the post-test and apply what you have learned.
6. Enjoy studying!

3. Objectives:
1. differentiate equations and inequalities;
2. illustrate linear inequalities in two variables;
3. determine the solutions of linear inequality in two
variables;
4. graph linear inequalities in two variables in slope-
intercept and standard form: and
5. solve problems involving linear inequalities in two
variables.

4. Pre-Test:
Choose the letter of the correct answer. Write the chosen letter on a
separate sheet of paper.
1. A mathematical sentence in a form of : Ax + By < C, Ax + By ≤ C,
Ax +By> C, Ax + By ≥, where A, B and C are real numbers and A & B
cannot be both equal to zero is called ______.
A. Linear Expression C. Linear Inequality
B. Linear Equation D. Linear Inequality in Two Variables
2. Which of the following is not an example of linear inequality in two
variables?
A. y > 2x -1 C. 3x – y = 1
B. a + b ≥ 5 D. 5e < 3 + f 3.

5. 3. The boundary line of the graph of linear inequality in two
variables is also called ________.
A. dashed line C. plane divider
B. half-plane D. solid line
4. Which of the following ordered pairs is a solution of the inequality
2x + 6y ≤ 10?
A. (3, 1) C. (1, 2)
B. (2, 2) D. (1, 0)
5. What linear inequality is represented by the graph at the right?
A. x - y > -2 C. x - y > 2
B. x + y > -2 D. x + y > 2

6. Review: Insert any object/s based on inequality
The use of inequality signs:
< ≥
> ≤
=

7. Introduction

8. LESSON 1
THE DIFFERENCE OF LINEAR
EQUATION IN TWO VARIABLES
AND LINEAR INEQUALITY
IN TWO VARIABLES

9. Definition
Linear Equation vs Linear Inequality
A mathematical sentence in the form
Ax+By+C < 0 Ax+By+C > 0
Ax+By+C ≤ 0 Ax+By+C ≥ 0
Where A, B, and C are real numbers and
A & B cannot be both equal to zero is
called a linear inequality in two
variables namely x & y.
An equation that can be put
in the form
Ax+By+C=0
where A,B, and C are real
numbers and A & B cannot be
both equal to zero is called a
linear equation in two
variables namely x & y.

11. linear equation
1
linear inequality
1
2x -4y+6 > 0
y ≤ 2x – 1
x ≥ 2 (shorthand
for x + 0y ≥ 2)
y <10 (shorthand
for 0x + y < 10)
2x -4y+6 = 0
y = 2x – 1
x = 8 (shorthand
for x + 0y = 8)
y = 15 (shorthand
for 0x + y = 15

12. y = x + 1 y < x +1
The Solutions
are many points
found in the line.
The Solutions
are any points in
the shaded area
Note: If the symbol
use is > or < , use
dashed line in
graphing

13. A. x = 2 & y = 3
B. x = 1 & y = 2
C. x = -4 & y = -3
E. x = 3 & y = 1
F. x = 0 & y = -3
G. x = -5 & y = - 6
Note: If the symbol used is ≤ or ≥ , we
used solid line ______ in graphing.
Y ≤ x + 1

14. A. x = 3 & y = 1
B. x = 0 & y = -3
C. x = -5 & y = - 6
The Solutions are any
points found ON THE
LINE and IN THE
SHADED AREA
Solution:
Example:
LINEAR
INEQUALITY
LINEAR
EQUATION

15. Activity
The boat is sinking
– The boat is sinking, group the mathematical
statements whether it’s a linear equation or
inequality in two variables. Write only the
corresponding letter of your answer.
Linear Inequalities in
two variables
Linear Equations
in two variables

17. LESSON 2
LINEAR INEQUALITY
IN TWO VARIABLES

18. Definition
A mathematical sentence in a form of:
Ax + By < C, Ax + By > C,
Ax + By ≤ C, Ax + By ≥ C ,
where A, B and C are real numbers and
A & B cannot be both equal to zero is called
a linear inequality in two variables namely x
and y.

19. Just like linear equation in two variables, linear
inequality in two variables also have forms:
General Form: Ax + By + C < 0
Standard Form: Ax + By < C
Slope-Intercept Form: y < mx + b

20. Let’s learn how to read some inequality
sentences!
5x+3y>1
“5x plus 3y is greater than 1”
5x+3y<1
“5x plus 3y is less than 1”
5x+3y≤1
“5x plus 3y is less than or equal to 1”
5x+3y≥1
“5x plus 3y is greater than or equal to 1”

21. GRAPH OF THE LINEAR
INEQUALITY IN TWO VARIABLES
We are not going to graph yet, this is to describe
what linear inequality in two variables graph looks like.
A linear inequality divides the coordinate plane
into two halves by a boundary line.
The plane separated by the boundary line is called
half-planes.
This boundary line is also known as plane divider.

22. The plane divider may be dashed or solid line.
If the symbol is > or < , the boundary line is dashed.
If the symbol is ≥ or ≤ , the boundary line is solid.
Plane Divider or
Boundary Line

23. 3x + 4y < 10 3x + 4y ≤ 10
The graph has shaded area. It represents the
solutions.
Dashed line is used to show that the points lie on the
boundary line are NOT part of the solution.
Solid line is used to show that the points lie on the
boundary line are part of the solution.

24. SOLUTIONS OF LINEAR
INEQUALITIES IN TWO
VARIABLES

25. The solution of a linear inequality in two
variables like Ax + By > C is an ordered pair
(x, y) that produces a true statement when
the values of x and y are substituted into the
inequality. Observe that ‘a solution’ is an
ordered pair. Thus, a solution consists of two
numbers.

26. Example 1:
Is (1,2) a solution to inequality
4x+3y>2?
To determine if (1, 2) is a solution to inequality 4x + 3y > 2 you
have to do the substitution.
ANSWER:
(1,2) is a solution of
4x + 3y > 2.
.
In (1,2), x = 1 and y = 2
4x + 3y > 2
Substitute: 4(1) + 3(2) > 2
Simplify 4 + 6 > 2
10 > 2 True
In this next example, notice that we are still using the
same inequality but different ordered pair.

27. Example 2.
Is (1, -1) a solution to inequality 4x+3y > 2 ?
Perform the substitution:
In (1, -1), x = 1 and y = -1
4x + 3y > 2
substitute: 4(1) + 3(-1) > 2
simplify: 4 - 3 > 2
1 > 2 FALSE
ANSWER:
(1,-1) is NOT a
solution of 4x + 3y > 2

28. IDENTIFYING THE SOLUTIONS OF
THE LINEAR INEQUALITIES IN TWO
VARIABLES USING ITS GRAPH

29. The graph of an inequality in two
variables is the set of points that
represents all solutions to the
inequality. Using the graph we can
determine if the coordinate is a
solution to the inequality or not.

30. Example 1: 2x – y < 5
Answer:
All the points in the shaded
region are solutions to the
inequality 2x – y < 5.
e.g. A (2,1) C (-3,2) F (0,-3)
All the points outside the
shaded region are not solutions.
e.g. D(3,-5) E (6,0)
The dashed line indicates that
all the points lie on the
boundary line are not solutions.
e.g. B (4,3) G (2,-1)

31. Activities:
Activity 2.1: The Label
Copy the figure below and use the following terms to label the graph.
plane
divider
x - axis
half-
plane
y - axis
half-
plane
the area
of
solutions
#2
#1

33. Activity 2.2:
Determine if the ordered pair satisfies the inequality. Tell whether
it’s A SOLUTION or NOT A SOLUTION.
Example: 1 – y > x ; (12,-9)
x, y
substitute: 1 – y > x
1 – (-9) > 12
simplify: 1 + 10 > 12
11 > 12 FALSE :
ANSWER: not a solution
1. x + y > 6; (5, 1)
2. x – y < 10; ( 4, -3)
3. y ≥ 2x + 5; (1, 7)
4. 2x > y – 4; (1.5, 6)
5. 8 + y ≤ 3x; (3, 4)

34. Activity 2.2:
B. Tell which of the given ordered pairs satisfy the inequality. Write
only the corresponding letter of your answer.
.
NOTE: The answer on each item could be more than one

35. Activity 2.2:
B. Tell which of the given ordered pairs satisfy the inequality. Write
only the corresponding letter of your answer.
.
NOTE: The answer on each item could be more than one

36. Activity 2.2:
B. Tell which of the given ordered pairs satisfy the inequality. Write
only the corresponding letter of your answer.
.
NOTE: The answer on each item could be more than one

37. Activity 2.2:
B. Tell which of the given ordered pairs satisfy the inequality. Write
only the corresponding letter of your answer.
.
NOTE: The answer on each item could be more than one

38. Activity 2.2:
B. Tell which of the given ordered pairs satisfy the inequality. Write
only the corresponding letter of your answer.
.
NOTE: The answer on each item could be more than one

39. Activity 2.2:
B. Tell which of the given ordered pairs satisfy the inequality. Write
only the corresponding letter of your answer.
.
NOTE: The answer on each item could be more than one

40. LESSON 3
GRAPHING LINEAR INEQUALITY
IN TWO VARIABLES

41. LINEAR EQUATION LINEAR INEQUALITY
When the equal sign in a linear equation is replaced with an
inequality sign, a linear inequality is formed.
y < 3x + 1
The solution consists of all
points (x,y) that satisfy the
inequality.
The solution forms a region or
half plane with the line as the
border.
y = 3x + 1
The solution consist of all points
(x, y) that satisfy the equation.
The solution forms a line.

42.  Graph the boundary line of the inequality as it was a line.
If the inequality is < or > , make a dashed line.
If the inequality symbol is ≤ or ≥ , make a solid line.
 Test a point.
Choose a point (usually the origin (0,0), substitute it into inequality and
simplify to determine which half-plane is the region of solution
 Shade.
If the coordinates of the point (x,y) satisfy the original inequality (the
statement is TRUE), shade the same side of the line as the point.
If it is FALSE, then shade the other side of the line.
The shaded area represents the region of the solution.
STEPSIN GRAPHING LINEAR INEQUALITIES IN TWO VARIABLES

43. Example 1: Graph y > 2x – 5
Step 1. Graphthe boundary line of the inequalityas it was a line
The inequality y>2x–5 can be write as y=2x-5.
Remember how to graph an equation in y = mx+ b
1. Identify the slope and the y-intercept.
y = 2x – 5
slope(m) = 2 y-intercept(b) =-5

44. 2. Plot the
y=intercept
(b)=-5
followed by
the next
points using
the slope
(m=2 or 2/1)

45. 2. Connect the points
according to the inequality
sign used.
If the inequality is < or
>, make a a dashed line.
If the inequality symbol
is ≤ or ≥, make a solid
line.
In inequality y > 2x-5, the
sign is > (greater than),
thus we must use a dashed
line.

46. STEP 2. Test a point.It is advisable
to use the coordinates of the
origin (0,0) as it is way easier
to simplify.
(0,0) x=0 , y=0
y > 2x-5
Substitute (0) > 2(0) – 5
Simplify 0 > 0 – 5
0 > -5 TRUE
Why TRUE? Look at the number
line below. 0 is ABOVE -5. It
implies that 0 is larger number
than -5

47. Activity 3.1: Color Me.
Activity Acti
Shade the side of the plane divider where the solutions of
the inequality are found. Copy the graphs in your
notebook.
1. y < x + 4 2. y – x ≤ - 2 3. – x ≥ y – 3
4. x + y > 1 5. y < - 3

49. LESSON 4
SOLVING PROBLEMS INVOLVING
LINEAR INEQUALITY
IN TWO VARIABLES

51. Activity 4
Instruction: In your notebook, solve the following problems.
1. Mr. Tony bought 4kg. of soya beans (s) and 3 kg of brown sugar (b) and
paid of not more than Php 463. If a kilogram of brown sugar is Php 45,
how much is the cost of a kilogram of soya beans?
a. What are the given?
b. Write the mathematical statement.
c. How much is the cost of a kilogram of soya beans?
2. Jill wants to buy shorts (s) and t-shirts(t) and has less than P1000 to
spend. A pair of shorts is P200 and a t-shirt costs P100.
a. What are the given?
b. Write the mathematical statement.
c. Graph
d. If she’d buy 2 pairs of shorts, how many t-shirt could she have?

52. Thank you and God bless!