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G R A D E 8 M AT H E M AT I C S
LINEAR INEQUALITIES
IN TWO VARIABLES

CONTENT STANDARD:
The learner demonstrates understanding of key
concepts of linear inequalities in two variables.
Performance Standard:
The learner is able to formulate real-life problems
involving linear inequalities in two variables and solve
these with utmost accuracy using a variety of strategies.

Have you asked yourself how your
parents budget their income for your
family’s needs?
How students like you spend your
time studying, accomplishing school
requirements, surfing the internet, or
doing household chores?

mathematics 8 second quarter second module.pptx

 an inequality in which each side is a polynomial of degree 1 or
constant.
 A linear inequality in x and y can be written in one of the following
forms;
What is linear Inequality?
Where A, B and C are real numbers and A & B are both not equal to 0.

1. 2.
3. 4.
Example(s)
The solutions of linear Inequalities can be expressed in ordered pairs (x, y).
 An ordered pair (x,y) is a solution if it satisfies the linear inequality.

Directions: Supply each phrase with the most appropriate
word.
6. Less talk, more _______
7. More harvest, less _____
8. Less work, more ______
9. Less trees, more _______
10. More savings, less ____
1. Less money, more_______
2. More profit, less ________
3. More smile, less ________
4. Less make-up, more _____
5. More peaceful, less ______
PROBLEMS
INVESTMENT
WORRY
NATURAL
CHAOS
ACTION
FAMINE
RELAX
CLIMATE
CHANGE
EXPENSE

A linear inequality in two variables is an
inequality that can be written in one of the
following forms:
Ax + By < C Ax + By ≤ C
Ax + By > C Ax + By ≥ C
where A, B, and C are real numbers and A and B
are both not equal to zero.

EXAMPLE
TWO
VARIABLES
4 x – 3 y > 1
A B C
Where A, B, & C
are real numbers
and A & B are
both not equal
to zero.
Inequality
symbol

EXAMPLE
TWO
VARIABLE
S
3 x – 5 y ≤ 9
A B C
Where A, B, & C
are real numbers
and A & B are
both not equal
to zero.
Inequality
symbol

EXAMPLE
TWO
VARIABLE
S
2 x + 7 y ˂ 2
A B C
Where A, B, & C
are real numbers
and A & B are
both not equal
to zero.
Inequality
symbol

EXAMPLE
TWO
VARIABLES
8 x - 7 y ≥ 4
A B C
Where A, B, & C
are real numbers
and A & B are
both not equal
to zero.
Inequality
symbol

{𝑥+2 𝑦 >11
𝑦 ≤ 2 𝑥 −7
Problem 1:
Determine whether the ordered pair is a solution of
b
Equation 1
1st
: Substitute the value of the ordered pair (x, y)
2nd
: Use the rule of integers
Equation 2.
1st
: Substitute the value of the ordered pair (x, y)
2nd
: Use the rule of integers
Solution:
The ordered pair

These forms can also be written in slope-intercept
form of linear inequality as the following:
y < mx + b y ≤ mx + b
y > mx + b y ≥ mx + b,
where m is the slope and b is the y-intercept
One of the ways to determine the possible
solutions of a linear inequality in two variables is
through graphing.

For the inequality, the graph is a region or a half-
plane and the line defines the boundary of the
shaded region.
SHADED REGION represents the solution sets
of the linear inequality.This indicates that any
ordered pair in the shaded region serves as the
solutions.

If the inequality involves < or > ( 𝑖𝑠 𝑙𝑒𝑠𝑠
),
𝑡ℎ𝑎𝑛 𝑜𝑟 𝑖𝑠 𝑔𝑟𝑒𝑎𝑡𝑒𝑟 𝑡ℎ𝑎𝑛 the line drawn is
a dashed or broken line, which means the points
on the line are not included in the solution.
If the inequality involves ≤ 𝑜𝑟 ≥, ( is less than or
equal to or is greater than or equal to) the line
drawn is a solid line, this means that the points on
the line are included in the solution.

Consider the following graphs to better understand the
visual presentation of linear inequalities.
the line is a dashed or broken line.This means that the
points on this line are not included in the solution.

Consider the following graphs to better understand the
visual presentation of linear inequalities.
The line used is solid line.This means that the points on
the line are included in the solution

To graph linear inequality in two variables the following
steps are helpful.
Step 1: Transform the inequality into the slope-intercept
form.
Step 2: Get the slope and y-intercept
Step 3: Locate the y – intercept in the coordinate plane.
From it, plot the other points (at least two points) using
the slope.
Step 4: Connect at least two points to draw a line.

steps are helpful.
Note: (a) use dashed or broken line when the inequality
uses the symbols < and >.This means that the points on
this line are not included in the solution set. (b) use
solid line when the inequality uses the symbols ≥ and ≤.
This means that the points on this line are part of the
solution set.

steps are helpful.
Step 5: Notice that the line divided the plane into two.
To determine which half-plane will be shaded, take any
point from the half-plane. If the point (ordered pair)
satisfies the given inequality, shade the half-plane where
the point is located.
Step 6: Show the graph of the inequality.

Example #1: Graph 2x + 3y < 6
Step 1: Transform the inequality into slope-intercept form.
Step 2: Get the slope and y-intercept.
Step 3: Locate the y-intercept in the coordinate plane. From it,
locate the other point using the slope.
Step 4: Connect the two points by a line. Since the symbol
used is <, then use broken line.
Step 5: Determine which region or part to be shaded by
testing points below the line or above the line that satisfies
the given inequality.

Example #2: Graph 3x + 4y > 12
Step 4: Connect the two points by a line. Since the symbol
used is >, then use broken line.
Step 5: Determine which region or part to be shaded by
testing points below the line or above the line that satisfies
the given inequality.

Example #3: Graph 5 + 2 ≤ 0
𝑥 𝑦
Step 4: Connect the two points by a line. Since the symbol used is
≤, then use solid line.This means that all points on this line are
part of the solution set.
Step 5: Determine which region or part to be shaded by testing
points below the line or above the line that satisfies the given
inequality.

Example #3: Graph 2 + 5 ≥ 15
𝑥 𝑦
Step 4: Connect the two points by a line. Since the symbol used is
≥, then use solid line.This means that all points on this line are
part of the solution set.
Step 5: Determine which region or part to be shaded by testing
points below the line or above the line that satisfies the given
inequality.

The table below shows the summary of examples 1 to 4.
Notice that when the inequality symbol used is > or <, the
boundary line is broken line.When the inequality symbol used
is ≥ or ≤, the boundary line is solid line. In the examples
above, test points are used to determine whether the
solutions lie above or below the line.

The table below shows the summary of examples 1 to 4.
However, linear inequalities can also be graphed without using
test points provided that they are written in any of these
forms: < + , > + , ≤ + , or ≥
𝑦 𝑚𝑥 𝑏 𝑦 𝑚𝑥 𝑏 𝑦 𝑚𝑥 𝑏 𝑦 𝑚𝑥
+ .
𝑏

This can be presented in the table below:

Let’s Summarize
USINGTHE SLOPE ITS
INTERCEPT
How do you illustrate linear
inequality in two variables?

Directions: Tell which of the following is a linear
inequality in two variables.
1. 3x – y 12
≥ 6. -6x = 4 + 2y
2. 19 < y 7. x + 3y 7
≤
3. y = 25x 8. x > -8
4. x 2
≤ y + 5 9. 9(x – 2) < 15
5. 7(x - 3) < 4y 10. 13x + 6 < 10 – 7y

REFERENCE:
Grade 8 Teachers’ Guide, pp. 239-245
Grade 8 Learner’s Manual, pp. 216-222
USINGTHE SLOPE ITS
INTERCEPT

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mathematics 8 second quarter second module.pptx

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