We will discuss here about the laws of inequality.
I. 1. If m > n then (i) m + k > n + k
(ii) m – k > n – k, where k is any positive or negative number.
Example: If x + 5 > 25 then x + 5 + 3 > 25 + 3, i.e., x + 8 > 28.
If x + 3 > 12 then x + 3 – 3 > 12 – 3, i.e., x > 9
If 7x + 3 > 6x + 5 then 7x + 3 – 6x – 3 > 6x + 5 - 6x – 3, i.e., x > 2
2. If m > n then (i) km > kn, where k is a positive number
(ii) km < kn, where k is a negative number
Example: If 7x > 5 then \(\frac{1}{7}\) ∙ 7x > \(\frac{1}{7}\) ∙ 5, i.e., x > \(\frac{5}{7}\)
Note: If m > n then (-1)m < (-1) n, i.e., -m < - n.
3. If m > n then (i) \(\frac{m}{k}\) > \(\frac{n}{k}\), where k is a
positive number
(ii) \(\frac{m}{k}\) < \(\frac{n}{k}\), where k is a negative number.
Example: If 7x > 35 then \(\frac{7x}{7}\) > \(\frac{35}{7}\), i.e., x > 5.
If -5x > -20 then \(\frac{-5x}{-5}\) < \(\frac{-20}{-5}\), i.e., x < 4.
Similar law hold for the inequality “≥”.
II. 1. If m < n then m + k < n + k, m – k < n – k, where k is any number positive or negative.
2. If m < n then (i) km < kn, where k is a positive number.
(ii) km > kn, where k is a negative number.
3. If m < n then (i) \(\frac{m}{k}\) < \(\frac{n}{k}\), where k is a positive number
(ii) \(\frac{m}{k}\) > \(\frac{n}{k}\), where k is a negative number.
Note: If m < n then (-1) m > (-1) n, i.e., - m > - n
Similar law hold for the inequality “≤”.
III. 1. If mn > 0 then m > 0, n > 0 or m < 0, n < 0
2. If mn < 0 then m > 0, n < 0 or m < 0, n > 0.
Note: p < q and q > p are the same inequality.
p < q and q < r together is also written as p < q < r.
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