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


exp-inequality-01

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}.


exp-inequality-02

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