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6.7 Graph and Solving Quadratic Inequalities

6.7 quadratic inequalities

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Page 1: 6.7 quadratic inequalities

6.7 Graph and Solving Quadratic Inequalities

Page 2: 6.7 quadratic inequalities

Method ofGraph

sketching

Page 3: 6.7 quadratic inequalities

Forms of Quadratic InequalitiesForms of Quadratic Inequalitiesy<ax2+bx+c y>ax2+bx+cy≤ax2+bx+c y≥ax2+bx+c

Graphs will look like a parabola with a solid or dotted line and a shaded section.

The graph could be shaded inside the parabola or outside.

Page 4: 6.7 quadratic inequalities

Steps for graphingSteps for graphing

1. Sketch the parabola y=ax2+bx+c

(dotted line for < or >, solid line for ≤ or ≥)

** remember to use 5 points for the graph!

2. Choose a test point and see whether it is a solution of the inequality.

3. Shade the appropriate region.

(if the point is a solution, shade where the point is, if it’s not a solution, shade the other region)

Page 5: 6.7 quadratic inequalities

Example:Graph y ≤ x2+6x- 4

3)1(2

6

2

a

bx* Vertex: (-3,-13)

* Opens up, solid line

134189

4)3(6)3( 2

y 9- 5-

12- 4-

13- 3-

12- 2-

9- 1-

yx

•Test Point: (0,0)

0≤02+6(0)-4

0≤-4 So, shade where the point is NOT!

Test point

Page 6: 6.7 quadratic inequalities

Graph: y>-x2+4x-3

* Opens down, dotted line.

* Vertex: (2,1)2

)1(2

4

2

a

bx

1384

3)2(4)2(1 2

y

y

* Test point (0,0)

0>-02+4(0)-3

0>-3

x y

0 -3

1 0

2 1

3 0

4 -3

Test Point

Page 7: 6.7 quadratic inequalities

Last Example! Sketch the intersection of the given inequalities.1 y≥x2 and 2 y≤-x2+2x+4

Graph both on the same coordinate plane. The place where the shadings overlap is the solution.

Vertex of #1: (0,0)Other points: (-2,4), (-1,1),

(1,1), (2,4)

Vertex of #2: (1,5)Other points: (-1,1), (0,4), (2,4),

(3,1)

* Test point (1,0): doesn’t work in #1, works in #2.

SOLUTION!

Page 8: 6.7 quadratic inequalities

Solving Quadratic Inequalities

Page 9: 6.7 quadratic inequalities

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

Page 10: 6.7 quadratic inequalities

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.

Page 11: 6.7 quadratic inequalities

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

Page 12: 6.7 quadratic inequalities

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

Page 13: 6.7 quadratic inequalities

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

Page 14: 6.7 quadratic inequalities

From the sketch, we obtain the solution

3xor2x

Page 15: 6.7 quadratic inequalities

Graphical Solution:

0 2 3

Page 16: 6.7 quadratic inequalities

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 !!!

Page 17: 6.7 quadratic inequalities

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

Page 18: 6.7 quadratic inequalities

From the sketch, we obtain the solution

2 < x < 3

Page 19: 6.7 quadratic inequalities

0 2 3

Graphical Solution:

Page 20: 6.7 quadratic inequalities

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

Page 21: 6.7 quadratic inequalities

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

Page 22: 6.7 quadratic inequalities

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

Page 23: 6.7 quadratic inequalities

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