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.