A compound inequality is a connection or relation of two or more then two inequalities joined together with "and" or "or".

In an inequality "or" indicates that as long as either inequality is true, the entire system(sentence) is true. "Or" is the combination of union of the sets the individual statements.

"And" indicates that both statements of the compound inequality are true at the same time. It is the intersection or overlap of the solution sets for the individual statements.

Like x > 5 and x < 10 is a compound inequality, which states that x takes the values greater then 5 and less than 10.

0 $\leq$ x $\leq$ 4 is again a compound inequality, where x is either 0 or 5 or any number between 0 and 5.

Solving Compound Inequality with "And" and "Or":
To solve any compound inequality, we follow some steps:
(1) First solve each inequality separately.

(2) Add or subtract the number term on each side of both terms.

(3) Multiply or divide of each side of both inequalities, by the required number. If we
multiply any inequality by a number with the negative sign then sign of inequality
will change.

(4) We just remember the thing that the word "and" indicates the overlap or
intersection is the required result and the word "or" indicates combine the
solutions i.e. find the union of the solution sets of each inequality.

(5) Draw the solution on the number line if its required.

The graph of a compound inequality containing the word "and" is the intersection of the solution set of the two inequalities. A compound inequality divides the number line into three separate regions.

A compound inequality containing the word "and" is true iff both inequalities are true. For the graph of this type of inequality we draw two things

(a) One of the circles goes on the each number in the solution.

(b) A dark line is marked between the two circles.

A compound inequality containing the wore "or " is true if one or more of the inequalities are true. For graph of this, we remember following things:

(a) One of the circles goes on the each number in the solution.

(b) We mark a dark line in the direction indicated by the symbol with the number.

(c) If the dark line are going to each other, the answer is "all real numbers".

(d) If both dark line are going to the right, then the solution is the set of all number
> or $\geq$ the smallest value.

(e) If both lines are going to the left, then the answer is the set of all number < or
$\leq$ the largest value.

Example 1:
Solve for x:
4x + 5 < 17 and x - 4 > -10.
Solution:
First solve each inequality separately, since the word "and" indicates the overlap or intersection is the desire result. So
4x + 5 < 17 and x - 4 > -10

Subtract 5 both sides in first inequality and add 4 both sides in the second inequality,
4x + 5 - 5 < 17 -5 and x - 4 + 4 > -10 + 4

or 4x < 12 and x > -6

Divided first inequality by 4 and second inequality remains same,
x < 3 and x > -6

Here x < 3 indicates all the numbers to he left sides of 3 and x > -6 indicates all the numbers to the right sides of -6.
So the intersection of these two is all the number between -6 and 3. The solution set is { x I x >-6 and x < 3 }.
Graph of above:



Example 2:
Solve for x:
3x + 6 < -12 or -4x + 1 < 13
Solution: First solve each inequality separately, the word "or" indicates combine the
answers i.e. find the union of the solution sets of each inequality.
3x + 6 < -12 or -4x + 1 < 13
Subtract 6 both sides in first inequality and subtract 1 both sides in the second inequality,

3x + 6 -6 < -12 -6 or -4x + 1 -1 < 13 -1

$\Rightarrow$ 3x < -18 or -4x < 12

Divided first inequality by 3 and second by -4( when we divided second inequality by -4, then the sign of inequality is changed).

$\frac{3x}{3}$ < $\frac{-18}{3}$ or $\frac{-4x}{-4}$ < $\frac{12}{-4}$

$\Rightarrow$ x < -6 or x > -3

Here x < -6 indicates all the numbers to the left of -6 and x > -3 indicates all the numbers to the right of -3. Hence the solution set is
{ x I x < -6 or x > -3}