The document discusses linear inequalities in two variables. It defines a linear inequality as similar to a linear equation, except the equals sign is replaced with an inequality symbol such as <, >, ≤, or ≥. Examples of linear inequalities in two variables are provided. The key steps for graphing linear inequalities are outlined, including changing the inequality to an equation to determine the boundary line, identifying the type of boundary line (solid or broken), using a test point to determine the shaded region, and graphing the final inequality with appropriate shading and boundary lines.

1. Mathematics
8
Quarter 2 Week 1

2. Most Essential Learning Competencies
 differentiates linear inequalities in two
variables from
linear equations in two variables.
 Illustrates and graphs linear inequalities in two
variables.
 solves problems involving linear inequalities in
two variables.

3. A linear
inequality
is almost the same as a linear equation, except
the equals sign is replaced with an
inequality symbol. It can be written in different
forms such as:

4. Linear Inequality
“ax + by is less than c”
“ax + by is greater than c”
“ax + by is less than or
equal to c”
“ax + by is greater than or
equal to c”
ax + by ≤ c ax + by ≥ c
ax + by < c
ax + by > c

5. Example 1:
x-y < 5

6. Example 2:
5x+4y > 20

7. Example 3:
2x+3y ≤ 6

8. Example 4:
x – y ≥ 7

9. Assessment
Card

10. Differentiate Linear
equations and Linear
Inequality

12. Example of Linear
EQUATION
1. 2x + 5y = 10
As you can see it forms a line

13. Example of Linear
EQUATION
2. 3x -4y =12
Again this example forms a
line

14. Example of Linear
INEQUALITY
1. 3x – y ≤ 9
It has a shaded region and a
solid line as a boundary.

15. Example of Linear
INEQUALITY
2. 2x - 3y > 60
It has a shaded region and a
solid line as a boundary.

16. Assessment
Card

17. Graphs Linear Inequalities in
Two Variables

18. The Graph of a Linear Inequality is the set of all
points that are solutions to the inequality.
Step 2: Solve for the boundary line of
the graph using x and y intercept
method.
Let y = 0 Let x = 0
x + y = 4 x + y = 4
x + 0 = 4 0 + y = 4
x = 4 y = 4
(4 ,0) (0,4)
Points (4,0) and (0,4) will be the
boundaries of the graph.
2
Step 1: Change the inequality
x + y < 4
into an equation x + y = 4.
1
Example 1:
x + y < 4

19. The Graph of a Linear Inequality

20. The Graph of a Linear Inequality
Step 4: Decide which part of the plane
will be shaded by performing test point
(0,0) to the inequality.
Substitute x = 0 and y= 0
to x + y < 4
x + y < 4
0 + 0 < 4
0 < 4 - TRUE
If proven TRUE, the half plane that
contains the origin will be shaded and if
it is FALSE, the origin must not be
included in shading.
4
Step 3: Determine what type
of line (solid or broken) will
be used. For symbols > or <
use broken lines. Therefore.
the two points on the boundary
line (4,0) and (0,4) will be
holes.
3

21. The Graph of a Linear Inequality
Step 5: Since 0 < 4 is TRUE,
the graph will be:
The origin (0,0) is included.
The shaded region contains all
the possible solutions to the
inequality x + y < 4.
5

22. The Graph of a Linear Inequality
Step 2: Solve for the boundary line of
the
graph using x and y intercept method.
Let y = 0 Let x = 0
8x – 6y = 24 8x – 6y = 24
8x – 6(0) = 24 8(0) - 6 y = 24
8x – 0 = 24 0 - 6y = 24
8x = 24 -6y = 24
x = 3 y = -4
(3,0) (0,-4)
Points (3,0) and (0,-4) will be the
boundaries of the graph.
2
Step 1: Change the inequality
8x – 6y ≥ 24 into equation
8x – 6y = 24.
1
Example 2:
8x – 6y ≥ 24

23. The Graph of a Linear Inequality

24. The Graph of a Linear Inequality
Step 4: Decide which part of the plane
will be shaded by performing test point
(0,0) to the inequality.
Substitute x = 0 and y= 0
to 8x – 6y ≥ 24
8(0) – 6(0) ≥ 24
0 - 0 ≥ 24
0 ≥ 24 – FALSE
If proven TRUE, the half-plane that
contains the origin will be
shaded and if it is FALSE, the origin
must not be included in
shading.
4
Step 3: Determine what type
of line (solid or broken) will
be
used. For symbols ≥ or ≤ use
a solid line. Therefore the
two points on the boundary
line (4,0) and (0,4) will be part
of the line.
3

25. The Graph of a Linear Inequality
Step 5: Since 0 ≥ 24 is FALSE, the
graph will
be:
The origin (0,0) is not
included.
The shaded region contains all the
possible solutions for the given
inequality
5

26. Assessment
Card

27. A Picture Always
Reinforces the
Concept
Images reveal large
amounts of data, so
remember: use an
image instead of long
texts

28. THANK
YOU!
Excited to be
BTS!
Back To School!