linear inequalities foundation part i. an inequality shows a relationship between two variables,...
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Linear Inequalities Foundation Part I
• An INEQUALITY shows a relationship between two variables, usually x & y
• Examples– y > 2x + 1 – y < x – 3– 3x2 + 4y ≥ 12
What is an inequality?
Objective of these next few slides
• Read a graph and write down the inequalities that contain a region
• Draw inequalities and indicate the region they describe
• You need to know how to plot straight line graphs (yesterday)
For e
xam
ple
x
yx > 2
X=2
When dealing with ONE inequality,we SHADE IN the REQUIRED REGION
For e
xam
ple
x
yx < -2
X=-2
For e
xam
ple
x
yy < -1
y=-1
For e
xam
ple
x
yy < 2x +1 y= 2x+1
Which sideis shaded?
Pick a pointNOT on line
(1,2)
Is 2 < 2 x 1 + 1 ?
YES (1,2) lies in the required region
For e
xam
ple
x
yy > 3x - 2 y= 3x-2
Which sideis shaded?
Pick a pointNOT on line
(2,1)
Is 1 > 3 x 2 - 2 ? NO
(2,1) does NOT lie in the required region
How
to d
raw
gra
ph o
f equ
atio
n
x
yy = 3x + 2
Shade IN the Region for y > 3x + 2
(2,1)Is 1 > 3 x 2 + 2 ?
NO
(2,1) does NOT lie in the required region
How
to d
raw
gra
ph o
f equ
atio
n
x
y4y + 3x = 12
Shade IN the Region for 4y + 3x > 12 (3,2)
Is 4 x 2 + 3 x 3 > 12 ?
YES (3,2) DOES lie in the required region
Regions enclosed by inequalities
y = 3
x = 4x + y = 4y < 3
x < 4
x + y > 4
(3,2)
2 < 3 ?
3 + 2 > 4 ?
3 < 4 ?
Part IIPart IISolving Linear and Solving Linear and
Quadratic InequalitiesQuadratic Inequalities
Linear Inequalities
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 :
Examples of linear inequalities:
123 x1. 2. xx 834
Dividing or multiplying by 1 gives 1 and 2BUT 1 is greater than 2
21 So,
21 We know ( 1 is less than 2 )
12
Linear Inequalities
These inequalities can be solved like linear equations EXCEPT that multiplying or dividing by a negative number reverses the inequality.
Examples of linear inequalities:
123 x1. 2. xx 834
Exercises
Find the range of values of x satisfying the following linear inequalities:
1. 3214 xx
2. 137 xx
Solution: 1324 xx 42 x
2 x
Solution: Either xx 317 x48
x 2Or 84 x 2 xDivide by -4:
2xso,
322 xxy
Quadratic Inequalities
Solution:
e.g.1 Find the range of values of x that satisfy 322 xx
Rearrange to get zero on one side: 0322 xx
0322 xx 0)3)(1( xx
1 x or 3x
322 xx is less than 0 below the x-axis
13 xThe corresponding x values are between -3 and 1
Let and solve 32)( 2 xxxf )(xfy
Method: ALWAYS use a sketch
542 xxy 542 xxy
Solution:
e.g.2 Find the values of x that satisfy 0542 xx
0542 xx 0)1)(5( xx
5 x or 1x
1 x
There are 2 sets of values of x
Find the zeros of where )(xf 54)( 2 xxxf
542 xx is greater than or
equal to 0 above the x-axis
5xor
These represent 2 separate intervals and CANNOT be combined
24 xxy
Solution:
e.g.3 Find the values of x that satisfy 04 2 xx
04 2 xx
0)4( xx
40 x
Find the zeros of where )(xf 24)( xxxf
24 xx is greater than 0
above the x-axis
This quadratic has a common
factor, x
or 4x0x
24 xxy
Be careful sketching this quadratic as the coefficient of is negative. The quadratic is “upside down”.
2x
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
1072 xxy 1072 xxy
Exercise
01072 xx 0)2)(5( xx
5 x or 2x
2 x
There are 2 sets of values of x which cannot be combined
1072 xx is greater than
or equal to 0 above the x-axis
5xor
1. Find the values of x that satisfy where 107)( 2 xxxf 0)( xf
Solution:
Now do Exercise 4A page 126