8.4 Solving Polynomial & Rational Inequalities

To solve a polynomial inequality:1) Write in standard form2) Find zeros3) Plot zeros on number line4) Choose test point in each interval to determine the sign

(factor, etc)

Ex 1) Solve 6x3 –10x2 < 24x

6x3 –10x2 – 24x < 0

2x(3x2 – 5x – 12) < 0

x = 05 25 4(3)( 12) 5 25 144 5 169

6 6 65 13 5 13 4

, 36 6 3

x

x x

want (–) values

0 3–2

– 48 – 40 + 48 –1–6 – 10 + 24

16 – 10 – 24

43

(–)

4384 – 160 – 96

(+) (–) (+)

43, (0,3) U

Ex 2) Solve x3 + 4x2 < x + 4

x3 + 4x2 – x – 4 < 0

x2(x + 4) – 1(x + 4) < 0

(x2 – 1)(x + 4) < 0

(x + 1)(x – 1)(x + 4) < 0

x = –1, 1, –4

–4 1

want (–) values

–5(–)(–)(–)

(–)

–2(–)(–)(+)

(+)

0(+)(–)(+)

(–)

(–∞, –4) (–1, 1)

–12

(+)(+)(+)(+)

Ex 3) Graph in the coordinate plane y > 3x3 – 5x + 1 Dotted line

Shade aboveUse the strategies from Ch 7!!

I.V.T.1

3 0 –5 13

2

3

3 6

–2 –1

7 15

–1

x

3 –3 –2 3

3 –6 7 –13

y

lower bound

–2

0

upperbound

I.V.T.

I.V.T.

Ex 4) Solve with the aid of a graph

Method 1

Find intersection pts

32 22 3x x Method 2

Find zeros

21

3 22

2

3

y x

y x

32 21 2 3y x x

Let’s have half of you try Method 1 and the other try Method 2

Where is parabola less than cube root?

[–1.6, 1.6]

Where is graph less than zero?

[–1.6, 1.6]

Homework

#804 Pg 409 #1, 6, 13, 18, 24, 29, 32, 35, 42, 43