Rational Inequalities Page 5

354

t

t#2:

Steps:

1. Get 0 on one side of inequality.

2. Get a common denominator and combine terms

3. Find Critical Values for numerator and denominator

4. Graph solutions on number line

5. Test each interval on number line

3 3

0354

t

t

01

354

t

t

t

t

0354

t

t

t

t

05

t

t

CV for Numerator

05t5t

CV for Denominator

0t

Plot CV on number line

5 0

Circle for this CV depends upon the inequality!

Circle for this CV is always open

Now we test each interval to determine where to shade! Just pick a number in the interval and test it in the original equation.

6 1 1

Find critical values by setting num and denom = 0

-6: 3.17>3 - TRUE

-1: -1>3 - FALSE

1: 9>3 - TRUE

5t OR 0t

21

53

a

a#3:

2 2

021

53

a

a

01

1

1

2

1

53

a

a

a

a

0

1

12

1

53

a

a

a

a

01

22

1

53

a

a

a

a

01

22

1

53

a

a

a

a

01

7

a

a

Find critical values by setting num and denom = 0

CV for Numerator

07a7a

CV for Denominator

01a1a

Page 5

Plot CV on number line

1 7

2 0 8

Now we test each interval to determine where to shade! Just pick a number in the interval and test it in the original equation.

-2: 11 2 - FALSE

0: -5 2 - TRUE

8:2.1 2 - FALSE

71 a

016

42

2

x

x#6:

Page 5

CV for Numerator

042 x 022 xx

2x 2x

CV for Denominator

0162 x 044 xx

4x 4x

Plot CV on number line

4 22 4

Circles for these CV’s depends upon the inequality!

Circles for these CV’s are always open

Now we test each interval to determine where to shade! Just pick a number in the interval and test it in the original equation.

5 3 0 3 5

-5: 2.3 0 - FALSE

-3: -.7 0 - TRUE

0: .25 0 - FALSE

3: -.7 0 - TRUE

5: 2.3 0 - FALSE

24 x OR 42 x

03

2

vv

#1:

CV for Numerator

02 v

0v

CV for Denominator

03v3v

Plot CV on number line

0 3

1 1 4

Open circle because of inequality

Always open circle b/c it’s CV for denom

-1: .5 < 0 - FALSE

1: -1 < 0 - TRUE

4: 8 < 0 - FALSE

30 v

Page 5

23

73

x

x#4:

2 2

023

73

x

x

03

3

1

2

3

73

x

x

x

x

03

62

3

73

x

x

x

x

03

1

x

x

03

1

x

x

CV for Numerator

01x1x

CV for Denominator

03x3x

Plot CV on number line

3 1

Open circle because of inequality

Always open circle b/c it’s CV for denom

4 2 0

-4: 5 > 2 - TRUE

-2: 1 > 2 - FALSE

0: 2.3 > 2 - TRUE

3x OR 1x

Page 5

016

22

x

x#5:

CV for Numerator

02x2x

CV for Denominator

Closed circle because of inequality

0162 x 044 xx

4x 4x

Always open circle b/c it’s CV for denom

Plot CV on number line

4 42

5 0 3 5

-5: -.8 0 - TRUE

0: .13 0 - FALSE

3: -.1 0 - TRUE

5: .33 0 - FALSE

4x OR 42 x

Page 5

Homework

• Page 6#2,6,10

34

2

n

n#2:

Page 6

3 3

034

2

n

n

04

4

1

3

4

2

n

n

n

n

04

123

4

2

n

n

n

n

04

142

n

n

CV for Numerator

0142 n142 n7n

Open circle because of inequality

CV for Denominator

04n4n

Plot CV on number line

7 4

Always open circle b/c it’s CV for denom

-8: 2.5 > 3 - FALSE

8 5 0

-5: 7 > 3 - TRUE

0: -.5 > 3 - FALSE

47 n

725

11

g

g#6:

Page 6

7 7

095

11

g

g

05

5

1

9

5

11

g

g

g

g

05

459

5

11

g

g

g

g

05

568

g

g

CV for Numerator

0568 g568 g7g

Open circle because of inequality

CV for Denominator

05g5g

Always open circle b/c it’s CV for denom

Plot CV on number line

7 5

8 6 0

-8: 4.3 < 7 - TRUE

-6: 15 < 7 - FALSE

0: -4.2 < 7 - TRUE

7g OR 5g

14

23

d

d#10:

1 1

014

23

d

d

04

4

1

1

4

23

d

d

d

d

04

4

4

23

d

d

d

d

04

22

d

d

CV for Numerator

022 d22 d1d

Closed circle because of inequality

CV for Denominator

04d4d

Always open circle b/c it’s CV for denom

Page 6

Plot CV on number line

4 1

5 0 2

-5: 13 1 - TRUE

0: .5 1 - FALSE

2: 1.3 1 - TRUE

4d OR 1d