The Chinese University of Hong Kong

EDD 5161R99EDD 5161R99

Group ProjectGroup Project

Chan Kwok Ping (S98118370)Seto Fung Mei (S98038260)

Form 5 --- lecturing

Learning PrerequisitesLearning Prerequisites::Sketching the graph of the corresponding

quadratic expressions.Method of factorization.Basic knowledge of Rectangular coordinate plane.

Students will be able to solve the quadratic inequalities by graphical method.

Represent the solutions graphically.

Aims and Objectives:Aims and Objectives:

ContentContent History (1) Sign of inequality

(2) Godfrey Harold Hardy

Inequality & Coordinate Plane

Solving Quadratic Inequality by Method of Graph Sketching

Exercise

Method ofGraph

sketching

Solve the quadratic inequality Solve the quadratic inequality xx2 2 – 5– 5x x + 6 > 0 graphically.+ 6 > 0 graphically.

Procedures:

Step (2): we have y = (x – 2)(x – 3) ,i.e. y = 0, when x = 2 or x = 3.

Factorize x2 – 5x + 6,

The corresponding quadratic function is y = x2 – 5x + 6

Sketch the graph of y = x2 – 5x + 6.

Step (1):

Step (3):

Step (4): Find the solution from the graph.

Sketch the graph Sketch the graph y =y = xx2 2 – 5– 5x x + 6 .+ 6 .

x

y

06 5

2 x x y

What is the solution of What is the solution of xx2 2 – 5– 5x x + 6 > + 6 > 0 0 ??

y = (x – 2)(x – 3) , y = 0, when x = 2 or x = 3.

2 3

above the x-axis.so we choose the portion

x

y

0

We need to solve x 2 – 5x + 6 > 0,

The portion of the graph above the x-axis represents y > 0 (i.e. x 2 – 5x + 6 > 0)

The portion of the graph below the x-axis represents y < 0 (i.e. x 2 – 5x + 6 < 0)

2 3

x

y

0

6 52

x x y

When x < 2x < 2,the curve is

above the x-axisi.e., y > 0

x2 – 5x + 6 > 0

When x > 3x > 3,the curve is

above the x-axisi.e., y > 0

x2 – 5x + 6 > 0

2 3

From the sketch, we obtain the solution

3xor2x

Graphical Solution:

0 2 3

Solve the quadratic inequality Solve the quadratic inequality xx2 2 – 5– 5xx + 6 < 0 graphically. + 6 < 0 graphically.

Same method as example 1 !!!Same method as example 1 !!!

x

y

0

6 52

x x yWhen 2 < x < 32 < x < 3,

the curve isbelow the x-axis

i.e., y < 0x2 – 5x + 6 < 0

2 3

From the sketch, we obtain the solution

2 < x < 3

0 2 3

Graphical Solution:

Solve

Exercise 1:

.012 xx

x < –2 or x > 1

Answer:

x

y

0

1 2 x x y

0–2 1

Find the x-intercepts of the Find the x-intercepts of the curve:curve:

(x + 2)(x – 1)=0(x + 2)(x – 1)=0

x = –2 or x = 1x = –2 or x = 1

–2 1

Solve

Exercise 2:

.0122 xx

–3 < x < 4

Answer:

x

y

0

122

x x y

0–3 4

Find the x-intercepts of the curve:Find the x-intercepts of the curve:

xx22 – x – 12 = 0 – x – 12 = 0

(x + 3)(x – 4)=0(x + 3)(x – 4)=0

x = –3 or x = 4x = –3 or x = 4

–3 4

Solve

Exercise 3:

.107

22

xx

–7 < x < 5

Solution:

x

y

0

35 22

x x y

0–7 5

Find the x-intercepts of the Find the x-intercepts of the curve:curve:

(x + 7)(x – 5)=0(x + 7)(x – 5)=0

x = –7 or x = 5x = –7 or x = 5

10

7

22

xx

271022 xx

03522 xx

057 xx–7 5

Solve

Exercise 4:

.3233 xxx

Solution:

x

y

0

35 22

x x y

Find the x-intercepts of the Find the x-intercepts of the curve:curve:

(x + 3)(3x – 2)=0(x + 3)(3x – 2)=0

x = –3 or x = 2/3x = –3 or x = 2/3

3233 xxx

03233 xxx

0233 xx

–3 23

0–3 23

x –3 or x 2/3

y

xoriginO x - axis

x P( x , y )

The horizontal number lineis called the x-axis.

The vertical number line is called the y-axis.

y - axis

The point of intersection of the axes is called the origin O.

y

Any point P on the plane is described by two numbers x and y called coordinates.

5

4

3

2

1

-4 -3 -2 -1 0 1 2 3 4

-1

-2

-3

-4

x

y

( 3, 1 )A

( 4, 3 )B( 1, 2 )C

( 2, 5 )D

What are the sign of the x- and y-coordinates of A,B,C and D?

(+,+)

5

4

3

2

1

-4 -3 -2 -1 0 1 2 3 4

-1

-2

-3

-4

x

y

(+,+)

E (4, 4 )

(3, 1 )F

(1, 2 )G

(2, 3 )H

What are the sign of the x- and y-coordinates of E,F,G and H?

(,+)

5

4

3

2

1

-4 -3 -2 -1 0 1 2 3 4

-1

-2

-3

-4

x

y

(,+) (+,+)

I (4,2 )

(3, 5 )L

(1, 3)K

(2,1)J

What are the sign of the x- and y-coordinates of I,J,K and L?

(,)

5

4

3

2

1

-4 -3 -2 -1 0 1 2 3 4

-1

-2

-3

-4

x

y

(,+) (+,+)

(,)

What are the sign of the x- and y-coordinates of M,N,P and Q?

( 3,5 )Q

( 4,3 )

( 1,4 )M

( 2,1 )NP (+,)

5

4

3

2

1

-4 -3 -2 -1 0 1 2 3 4

-1

-2

-3

-4

x

y

(,+) (+,+)

(,) (+,)

Shade the part that y>0y>0 (i.e.”++”).

5

4

3

2

1

-4 -3 -2 -1 0 1 2 3 4

-1

-2

-3

-4

x

y

(,+) (+,+)

(,) (+,)

Shade the part that y<0y<0 (i.e.””).

5

4

3

2

1

-4 -3 -2 -1 0 1 2 3 4

-1

-2

-3

-4

x

y

(,+) (+,+)

(,) (+,)

Shade the part that x>0x>0 (i.e.”++”).

5

4

3

2

1

-4 -3 -2 -1 0 1 2 3 4

-1

-2

-3

-4

x

y

(,+) (+,+)

(,) (+,)

Shade the part that x<0x<0 (i.e.””).

5

4

3

2

1

-4 -3 -2 -1 0 1 2 3 4

-1

-2

-3

-4

x

y

Shade the part that x<x<11.

5

4

3

2

1

-4 -3 -2 -1 0 1 2 3 4

-1

-2

-3

-4

x

y

Shade the part that x<2x<2.

5

4

3

2

1

-4 -3 -2 -1 0 1 2 3 4

-1

-2

-3

-4

x

y

Shade the part that x>1x>1.

5

4

3

2

1

-4 -3 -2 -1 0 1 2 3 4

-1

-2

-3

-4

x

y

Shade the part that 2<x<12<x<1.

5

4

3

2

1

-4 -3 -2 -1 0 1 2 3 4

-1

-2

-3

-4

x

y

Shade the part that 3<x<23<x<2.

5

4

3

2

1

-4 -3 -2 -1 0 1 2 3 4

-1

-2

-3

-4

x

y

Shade the part that xx<<2 or x>12 or x>1.

5

4

3

2

1

-4 -3 -2 -1 0 1 2 3 4

-1

-2

-3

-4

x

y

Shade the part that xx<<1 or x>41 or x>4.