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,

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

Note:
The order of an inequality is reversed 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,

\[\begin{gathered} {\text{x}} \geqslant {\text{a}} \Rightarrow – 8{\text{x}} \leqslant – 8{\text{a}} \\ {\text{x}} \geqslant {\text{a}} \Rightarrow – \frac{1}{8}{\text{x}} \leqslant – \frac{1}{8}{\text{a}} \\ \end{gathered} \]

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, and 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:
If ‘x’ are the real numbers that satisfy the linear inequality then this is how we graph them:

The solution of inequality $${\text{x}} > 3$$.
The graph of the solution of this inequality is given below.

The circle which marks 3 indicates that 3 is not included in the solution.

Now, the solution of inequality$${\text{x}} \geqslant 3$$ includes 3.
The graph of the solution of this inequality is given below.

The dot that marks 3 indicates that 3 is included in the solution of equality.