Math 8 Quarter 2 Week 1 LESSON: 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

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TheMathematician’sPrayer
Heavenly Father, thankYou for all the blessingsYou 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 toYou.
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!

3. MATHEMATICS 8
Quarter 2 Week 1
Linear
Inequalities
IN TWO VARIABLES
MR. CARLO JUSTINO J. LUNA
MALABANIAS INTEGRATED SCHOOL
Angeles City

4. FACEMath
L E A R N I N G M A T H I S F U N - T A S T I C
LEARNING
Competencies
4
» Differentiates linear
inequalities in two variables
from linear equations in two
variables (M8L-IIa-2)
» Illustrates and graphs linear
inequalities in two variables
» Solves problems involving
linear inequalities in two
variables (M8AL-IIa-4)

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Initial
Activity
5

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GROUP ME!
3𝑥 + 5𝑦 = 10 𝑦 > 2𝑥 − 7
4𝑥 ≤ 𝑦 − 1 9𝑥 − 10𝑦 = 8
−2𝑦 < 𝑥 + 3 5𝑦 ≥ 𝑥 + 2
𝑥 + 𝑦 = 12 𝑦 = 2𝑥 − 5
𝑥 + 5𝑦 = 10 3𝑥 > 2𝑦 + 1
GROUP A GROUP B

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GROUP A GROUP B
7
GROUP ME!
3𝑥 + 5𝑦 = 10 𝑦 > 2𝑥 − 7
4𝑥 ≤ 𝑦 − 1 9𝑥 − 10𝑦 = 8
−2𝑦 < 𝑥 + 3 5𝑦 ≥ 𝑥 + 2
𝑥 + 𝑦 = 12 𝑦 = 2𝑥 − 5
𝑥 + 5𝑦 = 10 3𝑥 > 2𝑦 + 1

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GROUP ME!
3𝑥 + 5𝑦 = 10 𝑦 > 2𝑥 − 7
4𝑥 ≤ 𝑦 − 1 9𝑥 − 10𝑦 = 8
−2𝑦 < 𝑥 + 3 5𝑦 ≥ 𝑥 + 2
𝑥 + 𝑦 = 12 𝑦 = 2𝑥 − 5
𝑥 + 5𝑦 = 10 3𝑥 > 2𝑦 + 1
GROUP A GROUP B
3𝑥 + 5𝑦 = 10
9𝑥 − 10𝑦 = 8
𝑥 + 𝑦 = 12
𝑦 = 2𝑥 − 5
𝑥 + 5𝑦 = 10
4𝑥 ≤ 𝑦 − 1
−2𝑦 < 𝑥 + 3
𝑦 > 2𝑥 − 7
5𝑦 ≥ 𝑥 + 2
3𝑥 > 2𝑦 + 1
LINEAR EQUATIONS LINEAR INEQUALITIES

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9
LINEAR INEQUALITIES IN TWO VARIABLES
3𝑥 + 5𝑦 = 10
9𝑥 − 10𝑦 = 8
𝑥 + 𝑦 = 12
𝑦 = 2𝑥 − 5
𝑥 + 5𝑦 = 10
4𝑥 ≤ 𝑦 − 1
−2𝑦 < 𝑥 + 3
𝑦 > 2𝑥 − 7
5𝑦 ≥ 𝑥 + 2
3𝑥 > 2𝑦 + 1
LINEAR EQUATIONS LINEAR INEQUALITIES
LINEAR
INEQUALITIES
< – less than
> – greater than
≤ – less than or equal to
≥ – greater than or equal to

10. FACEMath
L E A R N I N G M A T H I S F U N - T A S T I C
LINEAR INEQUALITIES
» Two linear expressions set that are
separated by symbols >, <, ≥, ≤
10
LINEAR INEQUALITIES IN TWO VARIABLES

11. FACEMath
L E A R N I N G M A T H I S F U N - T A S T I C
» 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
11
LINEAR INEQUALITIES IN TWO VARIABLES

12. FACEMath
L E A R N I N G M A T H I S F U N - T A S T I C
» 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
12
LINEAR INEQUALITIES IN TWO VARIABLES
4𝑥 ≤ 𝑦 − 1
−2𝑦 < 𝑥 + 3
𝑦 > 2𝑥 − 7
5𝑦 ≥ 𝑥 + 2
3𝑥 > 2𝑦 + 1
Examples:

13. FACEMath
L E A R N I N G M A T H I S F U N - T A S T I C
» An ordered pair (𝑥, 𝑦) is a
solution of a linear inequality in
two variables if aTRUE statement
results when the variables in the
inequality are replaced by the
coordinates of the ordered pair.
13
SOLUTION SET

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SOLUTIONS OF AN INEQUALITY
Determine whether each ordered pair is a solution of the given linear
inequality.
5𝑥 − 2𝑦 ≥ 14
1. (3, −2)
𝑥 𝑦
5𝑥 − 2𝑦 ≥ 14
5(3) −2(−2) ≥ 14
15 +4 ≥ 14
19 ≥ 14
TRUE
Therefore, (3, −2) is a solution
of the given linear inequality.
𝑥 𝑦
5(0) −2(4)
0 −8 ≥ 14
−8 ≥ 14
FALSE
Therefore, (0, 4) is not a solution
of the given linear inequality.
2. (0, 4)
5𝑥 − 2𝑦 ≥ 14
≥ 14

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SOLUTIONS OF AN INEQUALITY
Determine whether each ordered pair is a solution of the given linear
inequality.
𝑥 < 𝑦 + 2
𝑥 < 𝑦 + 2
0 < 4 + 2
0 < 6
TRUE
Therefore, (0, 4) is a solution
of the given linear inequality.
1. (0, 4)
𝑥 < 𝑦 + 2
7 < −5 + 2
7 < −3
FALSE
Therefore, (7, −5) is not a solution
of the given linear inequality.
2. (7, −5)
𝑥 < 𝑦 + 2
−5 < 1 + 2
−5 < 3
TRUE
Therefore, (−5, 1) is a solution
of the given linear inequality.
3. (−5, 1)

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SOLUTIONS OF AN INEQUALITY
» Clearly, a linear inequality can have two or more solutions.
» While the graph of 𝑥 + 𝑦 = 2 is a straight line, the graph of
𝑥 + 𝑦 > 2 is a half-plane.
Linear Equation Linear Inequality
The graph of a linear
inequality in two
variables is a HALF-
PLANE.
The shaded region
consists of the points
whose coordinates
satisfy the inequality.

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L E A R N I N G M A T H I S F U N - T A S T I C
How do we
graph linear
inequalities in
two variables?
17
Linear Inequality
The graph of a linear
inequality in two
variables is a HALF-
PLANE.
The shaded region
consists of the points
whose coordinates
satisfy the inequality.

18. GRAPHING
Linear
Inequalities

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StepsinGraphingLinearInequalitiesinTwoVariables
1. Transform the linear inequality 𝑎𝑥 + 𝑏𝑦 > 𝑐 into linear
equation 𝑎𝑥 + 𝑏𝑦 = 𝑐.
2. Determine the 𝑥- and 𝑦-intercept of the equation.
3. Plot and draw the boundary line. Use 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.
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 isTRUE, shade the side that
contains the test point. If the resulting inequality is
FALSE, shade the other side of the boundary line.

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Example 1: Graph 2𝑥 + 𝑦 ≤ 6.
2𝑥 + 𝑦 = 6
Rewrite the inequality as an equation.
2𝑥 + 𝑦 = 6
2𝑥 + 0 = 6
2𝑥 = 6
𝑥 = 3
𝑥-intercept is 3.
Solve and plot the intercepts.
2𝑥 + 𝑦 = 6
2 0 + 𝑦 = 6
𝑦 = 6
𝑦-intercept is 6.
The boundary is
a solid line since
the inequality
symbol is ≤.

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Example 1: Graph 2𝑥 + 𝑦 ≤ 6.
2𝑥 + 𝑦 ≤ 6
2 0 + 0 ≤ 6
0 ≤ 6
TRUE
Pick a test point. Use (0,0) if possible.
If the resulting
inequality is TRUE,
shade the side that
contains the test
point.

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Example 2: Graph 𝑥 < −5.
𝑥 = −5
Rewrite the inequality as an equation.
𝑥 = −5
𝑥-intercept is −5.
Solve and plot the intercepts.
There is no 𝑦-intercept.
The boundary is a
dashed line since the
inequality symbol is <.

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Example 2: Graph 𝑥 < −5.
𝑥 < −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.

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Asynchronous/Self-LearningActivities
Answer the following:
Q2W1LC1A:
• What I Know
• What’s More
• What I Can Do
Q2W1LC1B:
• What’s In
• What’s More (items 1 & 2 only)
Google Forms link
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25. MATHEMATICS 8
Quarter 2 Week 1
THANK
You
MR. CARLO JUSTINO J. LUNA
MALABANIAS INTEGRATED SCHOOL
Angeles City