Inequality is a statement as to which is larger or smaller of two quantities. The statements that ‘a’ is larger than ‘b’ is written a>b, and the consequential statement that b is smaller than a is b<a. inequality is shown by the sign > or <.

**Example:**

X +5 > 7

Y – 8< -3

The two inequality statements connected by disjunction or conjunction are known as compound inequality. Both inequality statements in the compound inequality is true when it is joined by the word ‘AND’ which means conjunction. Either statement is true when the compound inequality is disjunctive (‘Or’).

- Conjunction compound inequality:

In this type of compound inequality the ‘And ‘is used to join two inequality statements.

Example: 2x + 2 < 8 and 3x – 4 > -4

- Disjunction compound inequality:

In this type of compound inequality the ‘Or ‘is used to join two inequality statements.

Example: 2x - 3 > 7 or 3x – 2 < -5

- Double inequality:

Three sides in the inequality as in the form of a < bx + c < d is known as double inequality

Example: -22 < 5x -7 < 31

**Examples of compound inequality problems are:**

**Example 1:**

** ****Solve the inequality: x-3<7 and x+14>12**

**Solution:**

**Step 1 :** Solve the first inequality

x - 3<7 (Add 3 on both sides)

x < 10

**Step 2:** solve the second inequality

x + 14 >12 (Subtract 14 from both sides

x > -2

The final solution is:

**-2 < x < 10**

This means that all numbers between -2 and 10 are solutions

**Example 2:**

**Solve the compound inequality 3x + 5 < 1 or 3x > 2**

**Solution:**

**Step 1 :** Solve the first inequality

3x+5 < 1

Subtract 5 on both sides

3x+5-5 <1-5

3x < -4

Divide by 3 on both sides,

x < -4/3

**Step 2:** Solve the second inequality

3x > 2

Divide by 3 on both sides

x >2/3

The final solution is,

** -4/3 < x < 2/3**

**Example 3:**

** ****Solve the following compound double inequality**

**-2< 2x + 4 < 8**

**Solution:**

**Step 1:** Write as a compound inequality

-2 < 2x +4 and 2x + 4 < 8

**Step 2:** Subtract 4 on both sides for both inequalities

-6 < 2x and 2x < 4

**Step 3:** Divide by 2 on both sides for both inequalities

** -3 < x <2**

**Example 4:**

**-8 < 4 + x or 2 + 3x < 32**

**Solution:**

**Step 1:** Solving first inequality

-8 < 4 + x

Subtract 4 on both sides -8**-4**<4**-4**+x

-12 <x

**Step 2:** Solve the second inequality

2 + 3x < 32

Subtract 3 on both sides 2**-2** +3x< 32**-2**

3x<30

Divide by 3 on both sides**,**

x<10

The final solution is, **-12<x<10 .**

Practice Problem of compound inequality are:

1. -5 < 6x + 2 < 10

Answer: **-7/6<x<4/3**

2. -5 < 3x + 4 < 19

Answer: **-3<x <2**

3. 2x-3<5 and x+14>13

Answer: **-1<x<4**** **

4. 2(3 + x) > 4 or -3 - 2x ≤ 3

Answer: **-3 ≥ x > 1 .**