Basic Properties of Inequalities

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

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.

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.

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 1 , > 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

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.

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 > 5 + 4. True ∵ k > 5, ∴ k + 3 > 5 + 3. False ∵ k > 5, ∴ k – 1 > 5 – 1. ∵ k > 5, ∴ k – 10 > 5 – 10. False

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.

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 > 2 , 8 have we > 4 × Multiply both sides by 4. 8. 32 i.e. > 3, If 2. < b 3 , then < b 2 × Multiply both sides by 2. 6. 2 i.e. < b

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

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.

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

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

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.

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).