Inequality and Compound Inequality

Inequality:
An inequality expresses the relative order of two mathematical expressions. The symbols   < (less than),   \leqslant (less than or equal to),  > (greater than),  \geqslant (greater than or equal to) are used to write inequalities.
Note:
The sign of an inequality is unchanged if it is multiplied or divided by a positive number,
For example,

\begin{gathered} {\text{x}} < {\text{a}} \Rightarrow  3{\text{x}} < 3{\text{a}} \\ {\text{x}} < {\text{a}} \Rightarrow  \frac{1}{3}{\text{x}} < \frac{1}{3}{\text{a}} \\ \end{gathered}


Similarly,

\begin{gathered} {\text{x}} \geqslant {\text{a}} \Rightarrow  8{\text{x}} \geqslant 8{\text{a}} \\ {\text{x}} \geqslant {\text{a}} \Rightarrow  \frac{1}{8}{\text{x}} \geqslant \frac{1}{8}{\text{a}} \\ \end{gathered}

Note:
The order of an inequality is reserved if it is multiplied or divided by a. negative number.
For example,

\begin{gathered} {\text{x}} < {\text{a}} \Rightarrow  - 3{\text{x}} <  - 3{\text{a}} \\ {\text{x}} < {\text{a}} \Rightarrow  - \frac{1}{3}{\text{x}} <  - \frac{1}{3}{\text{a}} \\ \end{gathered}


Similarly,

\begin{gathered} {\text{x}} \geqslant {\text{a}}  \Rightarrow  - 8{\text{x}} \geqslant  - 8{\text{a}} \\ {\text{x}} \geqslant {\text{a}}  \Rightarrow  - \frac{1}{8}{\text{x}}  \geqslant  - \frac{1}{8}{\text{a}} \\ \end{gathered}

Linear Inequalities in One Variable:
Inequalities of the form {\text{ax}} + {\text{b}} < 0, {\text{ax}} + {\text{b}} \leqslant 0, {\text{ax}} + {\text{b}} > 0, {\text{ax}} + {\text{b}} \geqslant 0, where {\text{a}} \ne 0, b are constant, are called the linear equalities in one variable or first degree inequalities in one variable.
For example,

\begin{gathered} 2{\text{x}} + 9 < 7 \\ 3{\text{x}} + 7 < 2{\text{x}} - 9 \\ \end{gathered}


are all linear inequalities.

Compound Inequality:
A compound inequality is formed by joining two inequalities with a connective word such as “and” or “or”.
For example,{\text{x}} < 2 and 3{\text{x}} - 2 >  - 8 is a compound inequality.

Note:
The real numbers ‘x’ which satisfy the linear inequality in one variable ‘x’ from its solution.
For example,
The solution of inequality {\text{x}} > 3
The graph of the solution of this inequality is given below.


inequality-01

The hole marked on 3 indicates that 3 is not included in the solution.
Now, the solution of inequality{\text{x}} \geqslant 3 included 3
The graph of the solution of this inequality is given below.

inequality-02

The dark hole is marked on 3 indicating that 3 is included in the solution of equality.