Graphs of Inequalities in One Variable

Graphically, an inequality that involves only one variable $x$  is represented as points on a line:
e.g $-2\leq{x}<5$ can be represented on a number line as:


From the graph obtained, it cn be deduced that the solid or shaded circle indicates that the number $-2$ is included in
the range of $x$ while the open circle excludes the number $5$.
Referring back to sets theory, this is termed as a clopen set i.e a set that is both open and closed, as it can be seen
that the number $-2$ is on the close range while the number $5$ is on the open range.
Now lets take a look at linear inequality in one variable.

Linear Inequality in One Variable.
In this section, we take a look at linear inequalities in one variable

Example 1
Solve $24+5x\geq{2x-6}$ and represent the answer on a number line.

Solution
$24+5x\geq{2x-6}$
Collect like terms
$\Rightarrow{5x-2x}\geq{-6-24}$
$\Rightarrow{3x\geq{-30}}$
$\Rightarrow{x\geq{-10}}$
Below is the graph of the inequality:



Example 2
Solve the Inequality $\frac{3}{8}(x+1)+2>\frac{3}{4}(2x+1)+1$ and represent the answer on a number line.
Solution
$\Rightarrow{3(x+1)+2>\frac{3}{4}(2x+1)+1}$
Multiply both sides by $8$.
$\Rightarrow{3(x+1)+16>6(2x+1)+8}$
Expand the brackets
$\Rightarrow{3x+3+16>12x+6+8}$
Collect like terms
$\Rightarrow{3x-12x>14-19}$
$\Rightarrow{9x>-5}$
$\Rightarrow{x<-\frac{5}{9}}$
Below is the graph of the inequality: