1. Inequalities
Quadratic, Fractional
and
Irrational Inequalities

2. To solve more complicated inequalities,
use the following procedure:
• Make the RHS zero by shifting all terms to the LHS.
• Fully factorise the LHS to get critical values.
• Draw sign diagram for the LHS.
• Determine the values required from the sign diagram.

3. Quadratic Inequalities
Find the range of x for
making RHS zero
fully factorise LHS
find the critical values
draw sign diagram
-2
determine the
values

4. Find the range of values of x of the
following inequalities

5. Fractional Inequalities
Find the range of x for
making RHS zero
Simplify the equation
find the critical values
draw sign diagram
-3 4
determine the
values

6. Find the range of values of x of the
following inequalities

7. Irrational Inequalities
Find the range of x for
Square both sides
making RHS zero
Meeting the
requirement
draw sign diagram
Find common
solution

8. Find the range of values of x of the
following inequalities

9. Absolute Value Inequalities
x 0
Properties :
x a a x a
x a x a OR x a
2
x x

10. Example 1
Find the range of x for
Use property 1 3 x 2 3
3 2 x 3 2
Solution 1 x 5
Example 2
Find the range of x for 3x 6 3
Use property 2 3x 6 3 OR 3 x 6 3
3x 3 OR 3 x 9
Solution x 1 or x 3

11. Example 3
Find the range of x for x 2 1 2x
2 2
Square both sides (x 2) (1 2 x)
Simplify x
2
4x 4 1 4x 4x
2
2
3x 3 0
2
3( x 1) 0 ( 3)
Factorize ( x 1) ( x 1) 0
Find critical values x 1, x 1
Draw sign diagram
+ - +
-1 1
Solution x 1 OR x 1

12. Example 4
Find the range of x for x 1 3 x
2 2
Square both sides (x 1) (3 x)
2 2
Simplify x 2x 1 9 6x x
2x 6x 9 1
4x 8 (4)
Solution x 2
Meeting the requirement 3 x 0
x 3
x 3
Draw sign diagram
2 3 2 x 3

13. Find the range of values of x of the
following inequalities
1. 3 x 6 9 5. x 3 x 5
x 1 OR x 5 x 1
6. 2 x 1 x 2
2. 2 x 3 1
1
2 x 1 x OR x 3
3
2
3. 3 x 1 6 7. x 20 16
x 1 OR x 3 6 x 2 OR 2 x 6
2
4. x 3 x 2 8. x 3x 14 4
1
x 2 x 3 OR 2 x 5 OR x 6
2