# Examples of Inequality and Compound Inequality

Example:
Solve and graph the solution of the inequality ${\text{x}} + 3 > 4{\text{x}} + 6$

Solution:
We have
$\begin{gathered} {\text{x}} + 3 > 4{\text{x}} + 6 \\ \Rightarrow 3 – 6 > 4{\text{x}} – {\text{x}} \\ \Rightarrow – 3 > 3{\text{x}} \\ \Rightarrow \frac{{ – 3}}{3} > {\text{x}} \\ \Rightarrow {\text{x}} < – 1 \\ \end{gathered}$

Thus, the solution set is
Solution Set $= \left\{ {{\text{x:x}} \in \mathbb{R} \wedge {\text{x}} < – 1} \right\} = \left] { – \infty , – 1} \right[$

The graph of the solution set is

Example:
Solve and graph the solution of the inequality $11 – 2\left( {{\text{x}} – 1} \right) \geqslant 8 – 2\left( {{\text{x}} – 2} \right)$.

Solution:
We have
$\begin{gathered} 11 – 2\left( {{\text{x}} – 1} \right) \geqslant 8 – 2\left( {{\text{x}} – 2} \right) \\ \Rightarrow 11 – 2{\text{x}} + 2 \geqslant 8 – 2{\text{x}} + 4 \\ \Rightarrow 13 – 2{\text{x}} \geqslant 12 – 2{\text{x}} \\ \Rightarrow 13 \geqslant 12 \\ \end{gathered}$

Here equality is not possible because 13 is always greater than 12, in $13 > 12$. So, the solution of the given inequality is the set of all real numbers is $\mathbb{R}$.

The solution can be written as $\left] { – \infty ,\infty } \right[$