Math 8 Quarter 2 Week 2 LESSON: Systems of Linear Inequalities in Two Variables

1. WELCOME to our Math Class
LEARNING MATH IS
fun-tastic
MR. CARLO JUSTINO J. LUNA
MALABANIAS INTEGRATED SCHOOL
Angeles City

2. THE MATHEMATICIAN’S PRAYER
Heavenly Father, thank You for all the
blessings You gave unto us.
Add joy to the world;
Subtract evil from our lives,
Multiply the good things for us;
Divide the gifts and share them to others.
Convert badness to goodness.
Help us raise our needs to You.
Extract the roots of immoralities
and perform our different functions in life.
Tell us all that life is as easy as math.
Help us all to solve our problems.
These we ask in Jesus’ name,
the greatest mathematician
who ever lived on earth, Amen!
2

3. MATHEMATICS 8 | Quarter 2 Week 1
SYSTEMS OF LINEAR
INEQUALITIES
in Two Variables
MR. CARLO JUSTINO J. LUNA
MALABANIAS INTEGRATED SCHOOL
Angeles City

4. LEARNING
COMPETENCIES
M8AL-IIA-4
4
» Solve systems of linear
inequalities in two variables
» Graph systems of linear
inequalities in two variables

5. REVIEW
5

6. LINEAR INEQUALITIES
6
A linear inequality in two variables can be written in
four forms:
𝐴𝑥 + 𝐵𝑦 > 𝐶 𝐴𝑥 + 𝐵𝑦 ≥ 𝐶
𝐴𝑥 + 𝐵𝑦 < 𝐶 𝐴𝑥 + 𝐵𝑦 ≤ 𝐶
where 𝐴, 𝐵, and 𝐶 are real numbers, 𝐴 ≠ 0 and 𝐵 ≠ 0
4𝑥 > 𝑦 − 1
−2𝑦 < 𝑥 + 3
5𝑥 − 2𝑦 ≥ 14
𝑥 ≤ 𝑦 + 2

7. 7
» A linear inequality in
two variables has a
half-plane as the
set of solutions.
» A half plane is a
region containing all
points that has one
boundary, which is
a straight line that
continues in both
directions infinitely.
LINEAR INEQUALITIES

8. 8
» non-inclusive
» open
» dashed line
“INEQUALITY BROTHERS”
> < ≥ ≤
» inclusive
» closed
» solid line

9. 9
Steps to Graphing a Linear Inequality in Two Variables
1. Transform the linear inequality into linear equation.
2. Determine the x- and y-intercepts.
3. Plot the intercepts and draw the boundary line.
* If the inequality is inclusive (≤ and ≥), use a solid line.
* If the inequality is non-inclusive (< and >), use a dashed line.
4. Use the given inequality to choose a test point to be substituted in the
given inequality in order to identify the shaded side of the boundary line. It
is advisable to use (0, 0) for easy substitution of values.
5. If the resulting inequality is TRUE, shade the side that contains the test
point. If the resulting inequality is FALSE, shade the other side of the
boundary line.

10. SYSTEM OF
LINEAR
INEQUALITIES
IN TWO
VARIABLES
10

11. SYSTEM OF LINEAR INEQUALITIES
11
» If two or more linear inequalities are solved
simultaneously, this set of inequalities is called a
system of linear inequalities.
» The points that satisfy all the inequalities are the
solution of the system of linear inequalities.
2𝑥 − 𝑦 < 8
𝑥 + 𝑦 ≥ 7
3𝑥 + 𝑦 < 9
5𝑥 + 𝑦 < −5

12. SOLUTION OF SYSTEM OF LINEAR INEQUALITIES
12
Determine if the given point is a solution of the given inequality.
2𝑥 − 𝑦 < 8
𝑥 + 𝑦 ≥ 7
2𝑥 − 𝑦 < 8
a. (3, 4)
2(3) −(4) < 8
6 − 4 < 8
2 < 8
TRUE
𝑥 + 𝑦 ≥ 7
3 + 4≥ 7
7 ≥ 7
TRUE
(3, 4) is a solution of
the inequalities since it
satisfies the given
condition.

13. SOLUTION OF SYSTEM OF LINEAR INEQUALITIES
13
Determine if the given point is a solution of the given inequality.
2𝑥 − 𝑦 < 8
𝑥 + 𝑦 ≥ 7
2𝑥 − 𝑦 < 8
b. (2, −1)
2(2) −(−1)< 8
4 + 1 < 8
5 < 8
TRUE
𝑥 + 𝑦 ≥ 7
2 +(−1)≥ 7
2 − 1 ≥ 7
1 ≥ 7
FALS
E
(2, −1) is not a
solution of the
inequalities since one
of the inequalities did
not satisfy the given
condition.

14. SOLUTION OF SYSTEM OF LINEAR INEQUALITIES
14
Determine if the given point is a solution of the given inequality.
3𝑥 + 𝑦 < 9
5𝑥 + 𝑦 < −5
3𝑥 + 𝑦 < 9
a. (0, 0)
3(0) +(0) < 9
0 < 9
TRUE
5𝑥 + 𝑦 < −5
5(0)+ (0)< −5
0 < −5
FALS
E
(0, 0) is not a
solution of the
inequalities since
one of the
inequalities did not
satisfy the given
condition.

15. SOLUTION OF SYSTEM OF LINEAR INEQUALITIES
15
Determine if the given point is a solution of the given inequality.
3𝑥 + 𝑦 < 9
5𝑥 + 𝑦 < −5
3𝑥 + 𝑦 < 9
b. (−2, 3)
3(−2)+(3) < 9
−6 < 9
TRUE
5𝑥 + 𝑦 < −5
5(−2)+(3)< −5
−10 < −5
TRUE
+ 3
−3 < 9
+ 3
< −5
−7
(−2, 3) is a
solution of the
inequalities since it
satisfies the given
condition.

16. GRAPHING
SYSTEM OF
LINEAR
INEQUALITIES
IN TWO
VARIABLES
16

17. 17
STEPS IN GRAPHING
SYSTEM OF LINEAR INEQUALITIES
1. Change the inequality to equality symbol.
2. Identify the x and y intercepts of each equation.
3. Choose a test point to be substituted in the given
inequality in order to identify the shaded side of
the boundary line. It is advisable to use (0,0) for
easy substitution of values.
4. Graph the equation using dashed lines if the
inequality is < or >. However, graph using solid
lines if the inequality is ≤ or ≥, which means that
the points on the line are included in the solution
set.
5. Shade the region where the point belongs. Identify
the solution set. The points common to both are
the solutions of the system.

18. EXAMPLE 1:
18
Graph
2𝑥 − 𝑦 < 8
𝑥 + 𝑦 ≥ 7
𝟐𝒙 − 𝒚 = 𝟖
Rewrite the inequality as an equation.
2𝑥 − 𝑦 = 8
2𝑥 − 0 = 8
2𝑥 = 8
𝑥 = 4
𝑥-intercept is 4.
Solve and plot the intercepts.
2𝑥 − 𝑦 = 8
2(0) − 𝑦 = 8
−𝑦 = 8
𝑦 = −8
𝑦-intercept is −8.
The boundary is
a dashed line
since the
inequality symbol
is <.

19. EXAMPLE 1:
19
Graph
2𝑥 − 𝑦 < 8
𝑥 + 𝑦 ≥ 7
2𝑥 − 𝑦 < 8
2 0 − 0 < 8
0 < 8
TRUE
Pick a test point. Use (0,0) if possible.
If the resulting
inequality is TRUE,
shade the side that
contains the test
point.

20. EXAMPLE 1:
20
Graph
2𝑥 − 𝑦 < 8
𝑥 + 𝑦 ≥ 7
𝒙 + 𝒚 = 𝟕
Rewrite the inequality as an equation.
𝑥 + 𝑦 = 7
𝑥 + 0 = 7
𝑥 = 7
𝑥-intercept is 7.
Solve and plot the intercepts.
𝑥 + 𝑦 = 7
0 + 𝑦 = 7
𝑦 = 7
𝑦-intercept is 7.
The boundary is
a solid line since
the inequality
symbol is ≥.

21. EXAMPLE 1:
21
Graph
2𝑥 − 𝑦 < 8
𝑥 + 𝑦 ≥ 7
𝑥 + 𝑦 ≥ 7
0 + 0 ≥ 7
0 ≥ 7
FALSE
Pick a test point. Use (0,0) if possible.
If the resulting
inequality is FALSE,
shade the side that
does not contain
the test point.

22. EXAMPLE 1:
22
2𝑥 − 𝑦 < 8
𝑥 + 𝑦 ≥ 7
The points common
to both are the
solutions of the
system.
solutions

23. 23
EXAMPLE 2: Graph
3𝑥 + 𝑦 < 9
5𝑥 + 𝑦 < −5
𝟑𝒙 + 𝒚 = 𝟗
Rewrite the inequality as an equation.
3𝑥 + 𝑦 = 9
3𝑥 + 0 = 9
3𝑥 = 9
𝑥 = 3
𝑥-intercept is 3.
Solve and plot the intercepts.
3𝑥 + 𝑦 = 9
3 0 + 𝑦 = 9
𝑦 = 9
𝑦-intercept is 9.
The boundary is
a dashed line
since the
inequality symbol
is <.

24. 24
EXAMPLE 2: Graph
3𝑥 + 𝑦 < 9
5𝑥 + 𝑦 < −5
3𝑥 + 𝑦 < 9
3 0 + 0 < 9
0 < 9
TRUE
Pick a test point. Use (0,0) if possible.
If the resulting
inequality is TRUE,
shade the side that
contains the test
point.

25. 25
EXAMPLE 2: Graph
3𝑥 + 𝑦 < 9
5𝑥 + 𝑦 < −5
𝟓𝒙 + 𝒚 = −𝟓
Rewrite the inequality as an equation.
5𝑥 + 𝑦 = −5
5𝑥 + 0 = −5
5𝑥 = −5
𝑥 = −1
𝑥-intercept is −1.
Solve and plot the intercepts.
5𝑥 + 𝑦 = −5
5 0 + 𝑦 = −5
𝑦 = −5
𝑦-intercept is −5.
The boundary is
a dashed line
since the
inequality symbol
is <.

26. 26
EXAMPLE 2: Graph
3𝑥 + 𝑦 < 9
5𝑥 + 𝑦 < −5
5𝑥 + 𝑦 < −5
5 0 + 0 < −5
0 < −5
FALSE
Pick a test point. Use (0,0) if possible.
If the resulting
inequality is FALSE,
shade the side that
does not contain
the test point.

27. 27
EXAMPLE 2:
3𝑥 + 𝑦 < 9
5𝑥 + 𝑦 < −5
The points common
to both are the
solutions of the
system.
solutions

28. 28
ASYNCHRONOUS / SELF-
LEARNING ACTIVITIES
Answer the following:
Quarter 2 Week 2
• What’s In
• What’s More (items 1 & 3 only)
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29. 29
MATHEMATICS 8
Quarter 2 Week 2
THANK
You
MR. CARLO JUSTINO J. LUNA
MALABANIAS INTEGRATED SCHOOL
Angeles City