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