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Lucan Community College
Leaving Certificate
Mathematics
Higher Level
Mr Duffy
Linear and Quadratic
Inequalities
© Ciarán Duffy
Linear and Quadratic Inequalities
Linear Inequalities
Examples of linear inequalities:
1.
2. 4  3 x  8  x
3 x  12
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  2 ( 1 is less than 2 )
Dividing by 1 gives 1 and 2
BUT 1 is greater than 2
So,  1   2
2
1
0
1
2
Linear and Quadratic Inequalities
Linear Inequalities
Examples of linear inequalities:
1.
2. 4  3 x  8  x
3 x  12
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  2 ( 1 is less than 2 )
Dividing by 1 gives 1 and 2
BUT 1 is greater than 2
So,  1   2
The inequality has been reversed
Linear and Quadratic Inequalities
e.g.1. Find the values of x that satisfy the inequality
3 x  12
Solution: Divide by 3  x  4
e.g.2 Find the range of values of x that satisfy the
inequality 4  3 x  8  x
Solution: Collect the like terms
Notice the change
 x  3x  8  4
from “less than”
  4x  4
to “greater than”
Divide by 4: 
x  1
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
Linear and Quadratic Inequalities
Exercises
Find the range of values of x satisfying the following
linear inequalities:
1.
4x  1  2x  3
Solution: 4 x  2 x  3  1
2.

2 x  4

x  2
7  3x  x  1
Solution: Either 7  1  x  3 x  8  4 x
 2  x so, x  2
Or
 4 x  8 Divide by -4:  x  2
Linear and Quadratic Inequalities
Quadratic Inequalities
e.g.1 Find the range of values of x that satisfy
x  2x  3
2
Solution: Method: ALWAYS use a sketch
Rearrange to get zero on one side: x 2  2 x  3  0
Let f ( x )  x 2  2 x  3 and find the zeros of y  f ( x )
x 2  2 x  3  0  ( x  1)( x  3)  0
 x  1 or x  3
y  x2  2x  3
x 2  2 x  3 is less than 0 below the x-axis
The corresponding x values are between -3 and 1
 3 x 1
Linear and Quadratic Inequalities
e.g.2 Find the values of x that satisfy x 2  4 x  5  0
Solution:
Find the zeros of f ( x ) where f ( x )  x 2  4 x  5
x 2  4 x  5  0  ( x  5)( x  1)  0
 x5
or x  1
x 2  4 x  5 is greater than or
equal to 0 above the x-axis
There are 2 sets of values of x

x   1 or
x 5
These represent 2 separate intervals and
CANNOT be combined
yy  x 22  4 x  5
Linear and Quadratic Inequalities
e.g.3 Find the values of x that satisfy 4 x  x 2  0
Solution:
Find the zeros of f ( x ) where f ( x )  4 x  x 2
 4x  x2  0
 x (4  x )  0
This quadratic has a
common factor, x y  4 x  x 2
x  0 or x  4
than
sketching
this0quadratic as the coefficient of
4Be
x careful
x 2 is greater
2
negative.
x
aboveisthe
x-axis The quadratic is “upside down”.

0 x4
Linear and Quadratic Inequalities
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
Linear and Quadratic Inequalities
Exercise
1. Find the values of x that satisfy f ( x )  0 where
f ( x )  x 2  7 x  10
Solution:
x 2  7 x  10  0  ( x  5)( x  2)  0
 x  5 or
x2
y  x 2  7 x  10
x 2  7 x  10 is greater than
or equal to 0 above the
x-axis
There are 2 sets of values of x which cannot be
combined
 x  2 or
x 5
Linear and Quadratic Inequalities
Linear and Quadratic Inequalities
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.
Linear and Quadratic Inequalities
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
Linear and Quadratic Inequalities
Linear Inequalities
e.g.1. Find the values of that satisfy the inequality
3 x  12
Solution: Divide by 3  x  4
e.g.2 Find the range of values of x that satisfy the
inequality 4  3 x  8  x
Solution: Collect the like terms
Notice the change
 x  3x  8  4
from “less than”
  4x  4
to “greater than”
Divide by -4: 
x  1
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
Linear and Quadratic Inequalities
Quadratic Inequalities
e.g.1 Find the range of values of x that satisfy
x  2x  3
2
Solution: Method: ALWAYS use a sketch
Rearrange to get zero on one side: x 2  2 x  3  0
Let f ( x )  x 2  2 x  3 and find the zeros of y  f ( x )
x 2  2 x  3  0  ( x  1)( x  3)  0
 x  1 or x  3
y  x2  2x  3
x 2  2 x  3 is less than 0 below the x-axis
The corresponding x values are between -3 and 1
 3 x 1
Linear and Quadratic Inequalities
e.g.2 Find the values of x that satisfy x 2  4 x  5  0
Solution:
Find the zeros of f ( x ) where f ( x )  x 2  4 x  5
x 2  4 x  5  0  ( x  5)( x  1)  0
 x5
or x  1
y  x2  4x  5
x 2  4 x  5 is greater than or
equal to 0 above the x-axis
There are 2 sets of values of x

x   1 or
x 5
These represent 2 separate intervals and CANNOT be
combined