Laws of Inequality

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.

10th Grade Math

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