1. “Teach A Level Maths” Vol. 1: AS Core Modules
9: Linear and Quadratic Inequalities
© Christine Crisp

2. Quadratic Inequalities
e.g.1 Find the range of values of x that satisfy
Solution:
Method: ALWAYS use a sketch
Rearrange to get zero on one side:
Let and find the zeros of or
is less than 0 below the x-axis
The corresponding x values are between -3 and 1

3. These represent 2 separate intervals and CANNOT be combined
e.g.2 Find the values of x that satisfy
Solution:
Find the zeros of where or
is greater than or
equal to 0 above the x-axis
There are 2 sets of values of x or
These represent 2 separate intervals and CANNOT be combined

4. e.g.3 Find the values of x that satisfy
Solution:
Find the zeros of where
This quadratic has a common factor, x or
is greater than 0
above the x-axis
Be careful sketching this quadratic as the coefficient of is negative. The quadratic is “upside down”.

5. SUMMARY
Linear inequalities
Solve as for linear equations BUT
Keep the inequality sign throughout the working
If multiplying or dividing by a negative number, reverse the inequality
Quadratic ( or other ) Inequalities
rearrange to get zero on one side, find the zeros and sketch the function
Use the sketch to find the x-values satisfying the inequality
Don’t attempt to combine inequalities that describe 2 or more separate intervals

6. Exercise
1. Find the values of x that satisfy where
Solution: or
is greater than
or equal to 0 above the x-axis
There are 2 sets of values of x which cannot be combined or