1.

Warm-Up 3/31

C𝑉 𝑠=𝜋𝑟 2h=10

𝑉 𝑙=𝜋 (2𝑟 )2(1.5 h)

¿6𝜋 𝑟2 h¿6 ∙10

Rigor:You will learn how to solve polynomial inequalities and rational inequalities.

Relevance:You will be able to use polynomial inequalities

and rational inequalities to solve real world problems.

2-6 Nonlinear Inequalities

Polynomial Inequality has the form of:. is true when is negative. is true when is positive.

Example 1: Solve the inequality.

𝑥2−6 𝑥−30>−3

𝑥2−6 𝑥−27>0Let

𝑓 (𝑥 )=(𝑥+3)(𝑥− 9)

The solution set is. The graph supports the conclusion.

Example 2: Solve the inequality.

3 𝑥3− 4 𝑥2− 13𝑥−6 ≤ 0

Step 1 Find zeros. Let 𝑥=−1 , −

23

,∧3

The solution set is.

The graph supports the conclusion.

Step 2 Determine end behavior.Leading coefficient is positive degree is odd

Step 3 Use that there is a sign change at zeros to complete chart.

Example 3: Solve the inequality.

a. Step 1 Find zeros. Let

has no zeros.

The solution set is. The graph supports the conclusion.

b. The solution set is.

Step 2 Determine end behavior.

c. Step 1 Find zeros. Let

The solution set is.The graph supports the conclusion.d.

The solution set is.

Step 2 Determine end behavior.

V(– 2.5, 1.75) opens up

opens up

Rational Inequality must include the zeros of both the numerator and the denominator in the sign chart.

Example 4: Solve the inequality.4

𝑥−6+

2𝑥+1

>0

LCD: (x – 6)(x + 1)

4 (𝑥+1 )+2(𝑥−6)(𝑥− 6) (𝑥+1 )

>06 𝑥− 8

(𝑥− 6) (𝑥+1 )>0

Step 1 Find zeros and undefined points. Zero at

Let

The solution set is.

(𝑥+1)(𝑥+1)

∙4

𝑥−6+

(𝑥−6)(𝑥−6)

∙2

𝑥+1>0

√−1math!

2-6 Assignment: TX p145, 2-26 even