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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