# Linear Inequation in One Variable

We will discuss here about the linear inequation in one variable.

The mathematical statement which says that one quantity is not equal to another quantity is called an inequation.

For example: If m and n are two quantities such that m ≠ n; then any one of the following relations (conditions) will be true:

i.e., either (i) m > n

(ii) m ≥ n

(iii) m < n

Or, m ≤ n

Each of the four conditions, given above, is an inequation.

Consider the following statement:

“x is a number which when added to 2 gives a sum less than 6.”

The above sentence can be expressed as x + 2 < 6, where ‘<’ stands for “is less than”.

x + 2 < 6 is a linear inequation in one variable, x.

Clearly, any number less than 4 when added to 2 has a sum less than 6.

So, x is less than 4.

We say that the solutions of the inequation x + 2 < 6 are x < 4.

The form of a linear inequation in one variable is ax + b < c, where a, b and c are fixed numbers belonging to the set R.

If a, b and c are real numbers, then each of the following is called a linear inequation in one variable:

Similarly, ax + b > c             (‘>’ stands for “is greater than”)

ax + b ≥ c                           (‘≥’ stands for “is greater than or equal to”)

ax + b ≤ c                           (‘≤’ stands for “is less than or equal to”)

are linear inequation in one variable.

In an inequation, the signs ‘>’, ‘<’, ‘≥’ and ‘≤’ are called signs of inequality.

Let m and n be any two real numbers, then

1. m is less than n, written as m < n, if and only if n – m is positive. For example,

(i) 3 < 5, since 5 – 3 = 2 which is positive.

(ii) -5 < -2, since -2 – (- 5) = -2 + 5 = 3 which is positive.

(iii) $$\frac{2}{3}$$ < $$\frac{4}{5}$$, $$\frac{4}{5}$$ – $$\frac{2}{3}$$ = $$\frac{2}{15}$$ which is positive.

2. m is less than or equal to n, written as m ≤ n, if and only if n – m is either positive or zero. For example,

(i) -4 ≤ 7, since 7 – (-4) = 7 + 4 = 11 which is positive.

(ii) $$\frac{5}{8}$$ ≤ $$\frac{5}{8}$$, since $$\frac{5}{8}$$ - $$\frac{5}{8}$$ = 0.

3. m is greater than or equal to n, written as m ≥ n, if and only if m – n is either positive or zero. For example,

(i) 4 ≥ -6, since 4 – (-6) = 4 + 6 = 10 which is positive.

(ii) $$\frac{5}{8}$$ ≥ $$\frac{5}{8}$$, since $$\frac{5}{8}$$ – $$\frac{5}{8}$$ = 0.

4. m is greater than n, written as m > n, if and only if m – n is positive. For example,

(i) 5 > 3, since 5 – 3 = 2 which is positive.

(ii) -8 > -12, since -8 – (- 12) = -8 + 12 = 4 which is positive.

(iii) $$\frac{4}{5}$$ > $$\frac{2}{3}$$, since $$\frac{4}{5}$$ – $$\frac{2}{3}$$ = $$\frac{2}{15}$$ which is positive.