1. Basic Properties of Inequalities

2. Transitive Property : example For 1, 4 and 10 Since 1. > then 10
Also, if a < b and b < c,
then a < c. :
example
For 1, 4
and 10
Since 1.
>
then 10
> 4
> 1,
i.e.
> 1. 10 , 2
and If 2.
> - y x
then x
> 2 -
> y,
i.e.
> . y x

3. Transitive Property : example For , and 1 If 3. > r p 0, and If 4.
Also, if a < b and b < c,
then a < c. :
example
For ,
and 1 If 3.
> r p 0,
and If 4.
< v u 1
then
>
r . 0.
then
< u
The following are also true.
(a) If a b and b c, then a c.
(b) If a b and b c, then a c.

4. Follow-up question . then , and If 1. y x > . 2 then , and If 2. m
Fill in the blanks with appropriate inequality signs. .
then ,
and If 1. y x
> . 2
then ,
and If 2. m k
< 1.
then 1,
and If 3. n q
> .
then 4,
and 4 If 4. h g
>
∵ h > 4 and 4 > g,
∴ h > g, i.e. g < h.

5. Additive Property : example For Since 1. 5 1, > we have 5 1 , >
Also, if a < b, then
a + c < b + c. :
example
For
Since 1. 5 1,
>
we have 5 ,
> 2 +
Add 2 to both sides. 3 7
>
i.e. , 3 If 2. -
< y
3 ,
then -
< y 1 + 1 +
Add 1 to both sides. 2. 1
i.e. -
< + y

6. Additive Property For example : , If 3. > k h , If 4. < v u then
Also, if a < b, then
a + c < b + c.
For
example : , If 3.
> k h , If 4.
< v u
then
< v u , 2) ( - +
then
> k h . 4 + 2. 2
i.e. -
< v u
In fact, if a > b, then a – c > b – c.

7. Follow-up question
If k > 5, determine whether each of the following inequalities is true. 5 10 4. -
< k 1 4 3.
> 3 8 2. + 9 1.
True
∵ k > 5, ∴ k + 4 >
True
∵ k > 5, ∴ k + 3 >
False
∵ k > 5, ∴ k – 1 > 5 – 1.
∵ k > 5, ∴ k – 10 > 5 – 10.
False

8. Multiplicative Property
Also, if a < b and c > 0, then ac < bc; if a < b and c < 0, then ac > bc.
Please note the change of the inequality sign in the second expression.
Let’s consider the examples on the next page.

9. Multiplicative Property
Also, if a < b and c > 0, then ac < bc; if a < b and c < 0, then ac > bc. 2, 8
Since 1. :
example
For
> , 8
have we
> 4 ×
Multiply both sides by 4. 8. 32
i.e.
> 3, If 2.
< b ,
then
< b 2 ×
Multiply both sides by 2. 6. 2
i.e.
< b

10. Multiplicative Property
example
For : 4, If 3.
> h 4,
then h 2 × -
>
< 8. 2
i.e. -
< h 3, If 4. -
< u
3), (
then - u 5 × -
>
<
15. 5
i.e.
> - u

11. Note: , 1
then 0,
and If
(1) c b a ×
> .
i.e. c b a
> , 1
then 0,
and If
(2) c b a ×
<
> .
i.e. c b a
<
In conclusion,
for an inequality, if we multiply both sides by a negative number, we have to change the inequality sign.

12. Follow-up question 2 1. n m 2 4 - n 2. m 3 n 3. m -7 4. ¸ m -7 ¸ n
If m > n, fill in the blanks with appropriate inequality signs. 2 1. n m 2
∵ m > n, ∴ 2 × m > 2 × n. 4 - n 2. m
∵ m > n, ∴ –4 × m < –4 × n. 3 n 3. m
∵ m > n, ∴ m > n 3 1 ×
∵ m > n, ∴ m < n × -7 1 -7 4. m -7 n

13. Reciprocal Property : example For . 3 1 7 then < 3, 7 Since 1. >
Apply the multiplicative property to 7 > 3: . 3 1 7
then
< 3, 7
Since 1.
>
Since 7 > 3,
then 7  > 3  , 21 1 , 5 1
then -
> x 5, If 2. -
< x . 5 1 x
i.e. -
>
i.e > 3 1 7
< ∴ 7 1 3

14. Reciprocal Property In conclusion,
if we take the reciprocals of the numbers on both sides of an inequality, we have to change the inequality sign.

15. Follow-up question
If k > 3, fill in the blanks with appropriate inequality signs.
First take the reciprocals on both sides, then multiply both sides by (-1).