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.

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