Linear inequalities are defined as expressions in which two values are compared using the inequality symbols. An inequation is said to be linear if the exponent of each variable occurring in it is first degree only, and there is no term involving the product of the variables. Two real numbers or two algebraic expressions related by the symbol \(<\), \(>\), \(\leq\),  or \(\geq\) form an inequality.

Two real numbers or two algebraic expressions related by the symbol \(<\), \(>\), \(\leq\),  or \(\geq\) form an inequality.

\(3<5 ; 7>5\) are the examples of numerical inequalities while
\(x<5 ; y>2 ; x \geq 3, y \leq 4\) are some examples of literal inequalities.
\(3<5<7\) (read as 5 is greater than 3 and less than 7 ), \(3 \leq x<5\) (read as \(x\) is greater than or equal to 3 and less than 5) and \(2<y \leq 4\) are the examples of double inequalities.

Linear Inequality in One Variable

If \(a, b\), and \(c\) are real numbers, then each of the following is called a linear inequation in one variable:
1. \(a x+b>c\). Read as : \(a x+b\) is greater than \(c\).
2. \(a x+b<c\). Read as : \(a x+b\) is less than \(c\).
3. \(a x+b \geq c\). Read as : \(a x+b\) is greater than or equal to \(c\).
4. \(a x+b \leq c\). Read as : \(a x+b\) is less than or equal to \(c\).
In an inequation, the signs ‘ \(>{ }^{\prime},,^{\prime}<{ }^{\prime}, ‘ \geq^{\prime}\) and ‘ \(\leq\) ‘ are called signs of inequality.

Linear Inequations Example

Some examples of inequalities are given below:

\(
\begin{aligned}
& a x+b<0 \dots(1) \\
& a x+b>0 \dots(2) \\
& a x+b \leq 0 \dots(3) \\
& a x+b \geq 0 \dots(4) \\
& a x+b y<c \dots(5) \\
& a x+b y>c \dots(6) \\
& a x+b y \leq c \dots(7) \\
& a x+b y \geq c \dots(8) \\
& a x^2+b x+c \leq 0 \dots(9) \\
& a x^2+b x+c>0 \dots(10)
\end{aligned}
\)

Inequalities (1), (2), (5), (6) and (10) are strict inequalities while inequalities (3), (4), (7), (8), and (9) are slack inequalities. Inequalities from (1) to (4) are linear inequalities in one variable \(x\) when \(a \neq 0\), while inequalities from (5) to (8) are linear inequalities in two variables \(x\) and \(y\) when \(a \neq 0, b \neq 0\). Inequalities (9) and (10) are not linear (in fact, these are quadratic inequalities in one variable \(x\) when \(a \neq 0\) ).