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# Compound Inequalities

An equation with inequality symbol ($> ,\ < ,\ \geq ,\ or\ \leq$) instead of equality sign (=), is known as inequality. A compound inequality is an equation having two or more inequality signs. It is the joining of two simple inequalities usually by the conditions 'and' or ‘or’. It can also be defined as a set of equations joined together with two or more inequality symbols.

For example:
$x> -3\ and\ x< 5$, $a+5\geq 1\ or\ a-8\leq5$, $y-1< y\leq 4$ etc.

Solving Compound Inequalities

Compound inequalities are to be solved separately keeping following rules in mind

• The same number or expression can be added to or subtracted from both sides of an inequality. The value of the resulting inequality will be same as that of original one.
• If both sides of an inequality are multiplied or divided by the same positive number, then the value of resulting inequality will be same as the original one.
• Always flip the symbol of inequality while multiplying or dividing by a negative number. That means greater than will become less than and vice versa.
Notation of Compound Inequalities
There are two ways of notifying compound inequalities:
(1) Interval Notation: Compound inequalities can be denoted in form of union or intersection of open or closed intervals.
(2) Graphical Notation: Compound inequalities can also be denoted graphically on a number line.
For example: Compound Inequality $x\leq 8\ and\ x> -2$ defines values of x between -2 and 8 (including 8). This can be expressed as:
In interval notation: ( -2, 8 ].
In graphical notation:

Let us consider an example.

Example:
Solve the following inequalities: $2x+4> 16\ and\ 5x-7< -12$.
Solution:
$2x+4> 16\ and\ 5x-7< -12$
Solving both the equations separately -
$2x+4-4> 16-4\ and\ 5x-7+7< -12+7$
$2x> 12\ and\ 5x< -5$
$x> 6\ and\ x< -1$
which indicates that x has all the numbers greater than 6 and all the numbers less than -1. Therefore, interval notation: $(-\infty ,-1)\cup ( 6,\infty)$.

Graphical representation:

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