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

2. Module C1
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3. Examples of linear inequalities: 1. 2.
These inequalities can be solved like linear equations EXCEPT that multiplying or dividing by a negative number reverses the inequality.
Consider the numbers 1 and 2 :
We know
( 1 is less than 2 )
Dividing by -1 gives -1 and -2
BUT -1 is greater than -2
So, -2 -1 1 2

4. Linear Inequalities
Examples of linear inequalities: 1. 2.
These inequalities can be solved like linear equations EXCEPT that multiplying or dividing by a negative number reverses the inequality.
Consider the numbers 1 and 2 :
We know
( 1 is less than 2 )
Dividing by -1 gives -1 and -2
BUT -1 is greater than -2
So,
The inequality has been reversed

5. e.g.1. Find the values of that satisfy the inequality
Solution:
Divide by 3
e.g.2 Find the range of values of x that satisfy the inequality
Solution: Collect the like terms
Notice the change from “less than” to “greater than”
Divide by -4:
Tips:
Collecting the x-terms on the side which makes the coefficient positive avoids the need to divide by a negative number
Substitute one value of x as a check on the answer

6. Exercises
Find the range of values of x satisfying the following linear inequalities: 1.
Solution: 2.
Solution: Either
so, Or
Divide by -4:

7. 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

8. 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

9. 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”.

10. 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

11. 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

13. The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied.
For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.

14. 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
SUMMARY

15. e.g.1. Find the values of that satisfy the inequality
Divide by -4:
Solution:
Divide by 3
e.g.2 Find the range of values of x that satisfy the inequality
Solution: Collect the like terms
Notice the change from “less than” to “greater than”
Collecting the x-terms on the side which makes the coefficient positive avoids the need to divide by a negative number
Substitute one value of x as a check on the answer
Tips:
Linear Inequalities

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

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