This document discusses how to solve linear inequalities with one variable. It explains that a linear inequality is an open sentence with one variable and the first power, connected by a sign of inequality. It shows how to solve inequalities by addition, subtraction, multiplication, or division on both sides while maintaining the inequality sign. The properties of these operations on inequalities are described. Methods for determining the solution set of an inequality involving whole numbers are demonstrated.

1. I
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2. a cba
x + 4 = 10 x + 4 < 10 x + 4 > 10
1. Less than is written as “ < “
2. Greater than is written as “ > “
3. Less than or equal to is written as “≤ “
4. Greater than or equal to is written as “ ≥ “
The signs of inequality
Linear Inequality with one variable

3. What is Linear Inequality with One
Variable ?
Linear inequality with one variable is
an open sentence with one variable
and the first power which is
connected by sign of inequality
Linear inequality with one variable is
an open sentence with one variable
and the first power which is
connected by sign of inequality

4. How to solve these inequalities?
x + 4 < 10 x + 4 > 10

5. Solving A Linear Inequality with
One Variable
Sketching the graph of solution in a number line
Look at the following number line and then answer the question below!
What numbers are solutions of the inequality x < 3?
Is 4 a solution of that inequality?
Is 3 a solution of that inequality?
Is 2 a solution of that inequality?
Is 1 a solution of that inequality?
Is 0 a solution of that inequality?
Is -1 a solution of that inequality?
Is -2 a solution of that inequality?
Is -3 a solution of that inequality?
Can you
mention all
solutions of that
inequality?

6. Meanwhile, the solutions can be described on
the following number line.
The graph of solution of x ≤ 3 is
x = 3 on the line is not dotted
because 3 is not a solution
x = 3 on the graph is dotted because
3 is also a solution

7. Working out an Inequality by Addition or Subtraction
Look at statement -4 < 1. This statement is true. The number line below
shows what happens if 2 is added to both sides.
If 2 is added to both sides, then we obtain a statement -2 < 3. That
statement is also true.
In the example above, adding 2 to both sides does not change the truth
value of the statement.
+2 +2

8. Now, Look at statement -3 < 1. This statement is true. The number line
below shows what happens if 2 is substracted from both sides.
If 2 is subtracted from both sides, then we obtain a statement -5 < -1.
That statement is still true.
-2 -2
In the example above, subtracting 2 from both sides does not change the
truth of the statement.

9. Properties of addition or
subtracting in an inequality
• If a certain number is added to or subtracted
from both sides of an inequality, the symbol of
the inequality does not change, and the
solution does not change, either.
• The new linear inequality that we get if a
certain number is added to or subtracted from
both sides is called a linear inequality
equivalent to the original one.

10. 4 > -1
... = 4 × 2 ... -1 × 2 = ... (both sides are multiplied by 2)
... = 4 × 1 ... -1 × 1 = ... (both sides are multiplied by 1)
... = 4 × 0 ... -1 × 0 = ... (both sides are multiplied by 0)
... = 4 × -1 ... -1 × -1 = ... (both sides are multiplied by -1)
-8 = 4 × -2 ... -1 × -2 = ... (both sides are multiplied by -2)
-4 < -1
... = -4 × 2 ... -1 × 2 = ... (both sides are multiplied by 2)
... = -4 × 1 ... -1 × 1 = ... (both sides are multiplied by 1)
... = -4 × 0 ... -1 × 0 = ... (both sides are multiplied by 0)
... = -4 × -1 ... -1 × -1 = ... (both sides are multiplied by -1)
8 = -4 × -2 ... -1 × -2 = ... (both sides are multiplied by -2)

11. Working out Inequality by
Multiplication or Division
Consider the statement 4 > 1, 4 > -1, -4 <-1, 8 < 12, -8 < 12, and
the statement -8 > -12. Those two statements are true. Fill in the
blanks below. First fill it with a suitable number, and then fill it
with the sign “<“, “>”, or “=“.
4 > 1
8 = 4 × 2 ... 1 × 2 = ... (both sides are multiplied by 2)
... = 4 × 1 ... 1 × 1 = ... (both sides are multiplied by 1)
... = 4 × 0 ... 1 × 0 = ... (both sides are multiplied by 0)
... = 4 × -1 ... 1 × -1 = ... (both sides are multiplied by -1)
-8 = 4 × -2 ... 1 × -2 = ... (both sides are multiplied by -2)

12. 8 < 12
... = 8 : 4 ... 12 : 4 = ... (both sides are divided by 4)
4 = 8 : 2 ... 12 : 2 = ... (both sides are divided by 2)
... = 8 : ... 12 : = ... (both sides are divided by )
-8 = 8 : -1 ... 12 : -1 = -12 (both sides are divided by -1)
... = 8 : -2 ... 12 : -2 = ... (both sides are divided by -2)
-8 < 12
... = -8 : 4 ... 12 : 4 = ... (both sides are divided by 4)
-4 = -8 : 2 ... 12 : 2 = ... (both sides are divided by 2)
... = -8 : ... 12 : = ... (both sides are divided by )
8 = -8 : -1 ... 12 : -1 = -12 (both sides are divided by -1)
... = -8 : -2 ... 12 : -2 = ... (both sides are divided by -2)

13. Compare the sign in the box that you have filled with the sign
of the beginning statement. What happens if both sides are
multiplied by a positive number, by zero, or by negative
number? And what happens if both sides are divided by a
positive number , or by a negative number?
Compare the sign in the box that you have filled with the sign
of the beginning statement. What happens if both sides are
multiplied by a positive number, by zero, or by negative
number? And what happens if both sides are divided by a
positive number , or by a negative number?
-8 > -12
... = -8 : 4 ... -12 : 4 = ... (both sides are divided by 4)
-4 = -8 : 2 ... -12 : 2 = ... (both sides are divided by 2)
... = -8 : ... -12 : = ... (both sides are divided by )
8 = -8 : -1 ... -12 : -1 = 12 (both sides are divided by -1)
... = -8 : -2 ... -12 : -2 = ... (both sides are divided by -2)

14. ProPerties of multiPlication or
division on both sides of an inequality
For an inequality :
• If both sides are multiplied or divided by a
positive number (non zero), then the sign of
the inequality does not change.
• If both sides are multiplied or divided by a
positive number (non zero), then the sign of
the inequality changes into the opposite.

15. Determine the solution of inequality 3x – 1 < x + 3 with x variable of whole number set!
Method 1 :
1.By changing the sign of “<“ with “=“, we get the following equation 3x – 1 = x + 3.
2.Solve the equation.
3x – 1 = x + 3
2x = 4
x = 2
3.Take one counting number less than 2 and one counting number greater that 2.
For example 1 and 3.
4. Check which one meets 3x – 1 < x + 3.
• x = 1 then 3x – 1 < x + 3
3.(1) – 1 < 1 + 3
2 < 4
• x = 3 then 3x – 1 < x + 3
3.(3) – 1 < 3 + 3
8 < 6
So, the solution set of 3x – 1 < x + 3 is {x | x < 2 ; x is a member of whole
number}
Method 1 :
1.By changing the sign of “<“ with “=“, we get the following equation 3x – 1 = x + 3.
2.Solve the equation.
3x – 1 = x + 3
2x = 4
x = 2
3.Take one counting number less than 2 and one counting number greater that 2.
For example 1 and 3.
4. Check which one meets 3x – 1 < x + 3.
• x = 1 then 3x – 1 < x + 3
3.(1) – 1 < 1 + 3
2 < 4
• x = 3 then 3x – 1 < x + 3
3.(3) – 1 < 3 + 3
8 < 6
So, the solution set of 3x – 1 < x + 3 is {x | x < 2 ; x is a member of whole
number}

16. Method 2 :
3x – 1 < x + 3
⇔ 3x – 1 + 1 < x + 3 + 1
⇔ 3x < x + 4
⇔ 3x + (-x) < x + (-x) + 4
⇔ 2x < 4
⇔ 2x : 2 < 4 : 2
⇔ x < 2
Because x is element of whole number, then the value of x which conforms with x < 2 are x = 0
and x = 1 . So, the is solution set is {0,1}.
To, determine equivalent equations, you can also perform the following methods.
3x – 1 < x + 3
⇔ 3x – 1 + (-3) < x + 3 + (-3)
⇔ 3x -4 < x
⇔ 3x + (-3x) - 4 < x + (-3x)
⇔ -4 < -2x
⇔ -4 : -2 < -2x : -2
⇔ 2 > x
Since x is element of whole number, then the value of x which conforms with x < 2 are x = 0
and x = 1 . So, the is solution set is {0,1}. Using number line, chart of solution set is as shown in
figure below.
Both sides are added by 1Both sides are added by 1
Both sides are added by -xBoth sides are added by -x
Both sides are divided by 2Both sides are divided by 2
Both sides are added by -3Both sides are added by -3
Both sides are added by -3xBoth sides are added by -3x
Both sides are divided by -2, but
the sign of the inequality is
changed to >
Both sides are divided by -2, but
the sign of the inequality is
changed to >

17. Determine the solution of this inequality !
Solution by using method I:
⇔
⇔
⇔
⇔
⇔
⇔
⇔
⇔
⇔
Both sides are multiplied by 6Both sides are multiplied by 6
Both sides are subtractedd by 4Both sides are subtractedd by 4
Both sides are subtractedd by 9xBoth sides are subtractedd by 9x
Solving Inequality of Fraction FormSolving Inequality of Fraction Form
2
3
2)2(
3
1 x
x +>+
2
3
2)2(
3
1 x
x +>+
)
2
3
2(6)2(
3
1
6
x
x +>+×
xx 912)2(2 +>+
xx 91242 +>+
xx 9412442 +−>−+
xx 982 +>
xxxx 99892 −+>−
87 >− x
8
7
1
7
7
1
×−<−×− x
7
8
−<x
Both sides are multiplied byBoth sides are multiplied by 7
1
−

18. Solution by using method II:
⇔
⇔
⇔
⇔
⇔
⇔
⇔
⇔
This inequality can be solved by the other method
2
3
2)2(
3
1 x
x +>+
2
3
2
3
2
3
1 x
x +>+
3
2
2
3
2
3
2
3
2
3
1
−+>−+ xx
xx
2
3
3
1
1
3
1
+>
xxxx
2
3
2
3
3
1
1
2
3
3
1
−+>−
3
1
1
6
9
6
2
>− xx
3
1
1
6
7
>− x
3
4
7
6
)
6
7
(
7
6
×−<−×− x
7
8
−<x
Both sides are multiplied byBoth sides are multiplied by 7
6
−
Both sides are subtracted byBoth sides are subtracted by
3
2
−
Both sides are subtracted byBoth sides are subtracted by x
2
3
−