Linear inequalities in two variables can be represented as ax + by + c < 0, where a and b are not both zero and c can be any real number. The inequality symbol can be <, >, ≤, or ≥. Systems of linear inequalities in two variables can model real-life situations and be represented graphically by shading the region defined by the intersection of the inequalities. Solving problems involving linear inequalities in two variables involves reading the problem, representing it with inequalities, solving for the unknown quantity, and checking the solution.

2. A linear inequality in two variables is a sentence of the form
ax+by+c<0, a x + b y + c < 0 , where a and b are not both zero; c
can be any real number. The inequality symbol can be any of these: <
, > , ≤ , or ≥
3x + y > 20
2x - 3y < -4
8x + 10y ≥ 5
-5x + 12y ≤ 5
Linear Inequality

4. Linear
Inequalit
yin Two
Variables
Real – Life Situations
Linear
Equation
in Two
Variables
1. The sum of Sarah’s(s) and Lea’s(l) ages is at most 40.
2. According to the NSO, there are more female (f) Filipinos than
male(m) Filipinos.
3. The number of teachers(t) in our school is less than one-half of the
number of students(s) enrolled during this school year.
4. To get a passing mark in school, a student must have a grade (g)
of at least 75.
5. According to the DOH, the recovery rate(r) of the COVID-19
patients is greater than its fatality rate(f).
6. The difference of two consecutive integers is 82.
7. The total number of a cow’s (w) and a chicken’s(n) feet in a farm is
84.
8. The population(p) of the Philippines is at most 103, 000, 000.
9. The daily school allowance of Jillian(j) is less than the daily
allowance of Gwyneth(g).
10. Leo saved ₱160 in his piggy bank consisting of ₱1 coins and ₱5
coins.
ACTIVITY
s + l ≤ 40
f > m
Directions: Rewrite the following verbal phrases to its mathematical model on the
appropriate column that illustrates linear inequality or linear equations and answer
the questions that follow.

5. Activity
Directions: Supply the symbol for linear inequalities/ linear equations in two variables usingthe given terms
values of the variables. Substitute the value of the given variables in your inequality/equation to check if it
yields to a true statement. (x=2, y=4 and z= -5)
1. Given: 2x + 3y 40
Solution:
1. Given: 8y + 2z 22
Solution:
= 2x + y ___ 40
= 2(2) + 4___ 40
= 4 + 4___ 40
= 8 ___ 40
<
= 8y + 2z ___ 22
= 2x + 3y < 40
= 8(4) + 2(-5)___ 22
= 32+ -10___ 22
= 22___ 22
=
= 8y + 2z = 22

6. Activity 1a
Directions: Write the following linear inequalities in standard form.
1. Given: x + 3 > y
2. Given: x – 11 + 3y >
x + 3 > y
x > -3 + y
x -y > -3
x – 11 + 3y >
x + 3y > 11
Ax+ By < C
Ax+ By ≤ C
Ax+ By > C
Ax+ By ≥ C

7. Activity 1b
Directions: Determine if the given ordered pair is a solution of the
inequality.
1. Given: x + y > 8 (5,4)
Solution:
x y
x + y > 8
5 + 4 > 8
9 > 8 True
Conclusion: Solution
2. Given: 2x - 3y < 6 (2,-3)
x y
Solution:
2x - 3y < 6
2( ) – 3( )< 6
2 -3
4 + 9 < 6
13 < 6 False
Conclusion: Not Solution

8. Activity 1
Directions: Graph each of the following linear inequalities and use a test point to
verify theshaded region as its solution.
1. Given: 3x + 4y > 12
a. Change the inequality sign with equality sign
Solution:
3x + 4y 12
=
b. Give two values of x, then find the value of y
x = 0 x = 4
3x + 4y =12
3(0) + 4y =12
4y =12
4 4
y = 3
(0,3)
3x + 4y =12
3(4) + 4y =12
12+ 4y =12
4y =12-12
4y =0
4 4
y =0
(4,0)

9. c. Graph the two points generated on procedure b.
(0,3) (4,0) < , > dash line
≤ , ≥ solid line
d. Choose a testing point to identify
the solution, then shade it
(1,1)
3x + 4y > 12
3(1) + 4(1) > 12
3 + 4 > 12
7 > 12 False

12. Solves Problems Involving Linear Inequalities in Two Variables
Solving problems involving linear inequalities is
really difficult, so here are the suggested steps on
how to solve one:
Step 1. Read and analyze the problems.
Step 2. Represent the problem
through inequality.
Step 3. Solve for what is required.
Step 4. Check if necessary.

13. Example 1. Ashley bought five oranges and ten apples. The total amount that
she paid is at least Php 140. How much should an apple cost if an
orange cost ₱8?
Step 1. Determine the given and the required data.
Let x → no. of oranges
y → no. of apples
Step 2. Represent the problem through inequality
5x + 10y ≥ 140
Step 3. Solve for what is required:
5x + 10y ≥ 140
5(8) + 10y ≥ 140
40+ 10y ≥ 140
10y ≥ 140 -40
10y ≥ 100
10 10

14. 1. 2a > 3b +1
Solution:
2a > 3b +1
2(7) > 3(1) +1
14> 4
2. 6x < y +7
Solution:
6x < y +7
6(1) < 0 +7
6 < 7

15. 1. 3m – 8n > 13
Solution:
n = 4 find m = ?
3m – 8n > 13
3m – 8(4) > 13
3m – 32 > 13
3m > 13 + 32
3m > 45
3 3
m > 15

17. 1. It cost Laura ₱18 to make a face mask and ₱30 to make an improvised face shield. If
she intends to make 5 pieces of face mask, how many pieces of face shield can she
make if thetotal amount of production cost is less than ₱220?
Step 1. Determine the given and the required data.
Let x → cost face mask
y → cost face shield
Step 2. Represent the problem through inequality
18x + 30y < 220
Step 3. Solve for what is required:
90+ 30y < 220
30 30
18x + 30y < 220
18(5)+ 30y < 220
30y < 220 - 90
30y < 130

18. Solves Problems on Systems of Linear Inequalities in Two
Variables
Activity 1. PICK ME
• Given the graph of systems of
linear inequalities x < 3 and y >
-1. From the graphpick the
possible solutions from the
choices below that satisfy the
given inequalities below.
• Graph:
Choices:
A (0, -1) I (1, 2)
S (0, 0) E (2, 3)
H (2, 1) L (-2, 3)
B (-3, -1) F (-1,-2)
C (1, -2) D (0, 3)

20. 1. The Senior citizen’s organization must consist of 5 to 8 representatives
from the male and female group. The organization must include at least 2
males and 2 females.
a. write an inequality to represent the problem
b. graph the systems
Solution:
write an inequality to represent the problem
x – Male Senior
y – Female Senior
x ≥ 2
y ≥ 2

21. graph the systems
x ≥ 2
y ≥ 2
x = 2
(2,0)
y = 2
(0,2)
Test Point
(3,4)
x ≥ 2
3 ≥ 2 true
Test Point
(-5,4)
y ≥ 2
4 ≥ 2 true
Solution