Inequalities

Quadratic, Fractionaland

Irrational Inequalities

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.

Quadratic InequalitiesFind the range of x for

making RHS zero

fully factorise LHS

determine the values

draw sign diagram-2

find the critical values

Find the range of values of x of the following inequalities

Fractional InequalitiesFind the range of x for

making RHS zero

Simplify the equation

determine the values

draw sign diagram

-3

find the critical values

4

Find the range of values of x of the following inequalities

Irrational InequalitiesFind the range of x for

making RHS zero

Square both sides

Find common solution

draw sign diagram

Meeting the requirement

Find the range of values of x of the following inequalities

Absolute Value Inequalities

2xx

Properties :

axaax

axORaxax

0x

Example 1Find the range of x for

Solution

Use property 1

51 x

323 x

2323 x

Example 2Find the range of x for 363 x

363363 xORxUse property 2

Solution 31 xorx

9333 xORx

Example 3

Simplify

Square both sides

Find the range of x for

0)1()1( xx

xx 212

11 xORx

Factorize

Draw sign diagram

Solution

22 )21()2( xx

22 44144 xxxx 033 2 x

)3(0)1(3 2 x

Find critical values

-1 1

+ +-

1,1 xx

Example 4

Simplify

Square both sides

Find the range of x for

)4(84

1962

6912 22

x

xx

xxxx

xx 31

2xSolution

22 )3()1( xx

Meeting the requirement 03 x3 x3x

Draw sign diagram

2 3 32 x

Find the range of values of x of the following inequalities

963.1 x

132.2 x

613.3 x

23.4 xx

53.5 xx

212.6 xx

1620.7 2 x

4143.8 2 xx

51 xORx

12 x

31 xORx

2

12x

1x

33

1 xORx

6226 xORx

6523 xORxORx