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