P.o.D. – Identify any asymptotes. Vertical, Horizontal, and/or Slant

1.)

2.)

3.)

1.)VA: x= -3; SA: y=x+2

2.)VA: x=0; SA: y= -x+3

3.)VA: x= -4; SA: y = - ½ x + 1

2.7 – Nonlinear Inequalities

Learning Target: be able to solve polynomial inequalities; solve rational inequalities

EX: Solve

Factor the polynomial.

(x-6)(x+1)=0

X=6, x= -1

This divides the number line into three distinct regions.

All numbers less than -1.
X< -1 / All numbers between -1 and 6.
-1 < x < 6 / All numbers greater than 6.
X > 6

We must determine in which region the function is true.

Test Possible Values.

X= -4



False / X=0



True / x=8



False

Since the x=0 was true, our solution is all numbers between -1 and 6.

We can write this as a set (-1,6).

If this had been inclusive (, ), we would use brackets [a,b] as opposed to parentheses (a,b).

*We can also examine this problem graphically.

We are looking for where the graph is less than 0, or below the x-axis. This occurs between -1 and 6, just as we had previously solved.

EX: Solve .

We know that we should have three solutions because of the Fundamental Theorem of Algebra.

(Let’s solve this on the whiteboard).

Now test the intervals to determine the proper solution.

Set Notation

Again, we can confirm this using the graphing calculator.

EX: Solve .

First, determine the domain

.

Next, solve the equation

(Let’s do this work on the whiteboard).

X=1.

We now have 3 intervals.

All numbers less than or equal to 1 / All numbers between 1 (inclusive) and 3 / All numbers greater than 3.

Find the correct interval.

[1,3) – please note that the 3 is NOT inclusive because an asymptote occurs at that point.

Again, confirm this answer graphically.

. “When in DOUBT, GRAPH it out!”

Profit = Revenue – Cost

EX: The marketing department of a manufacturer has determined that the demand for a new product is p=500-.00005x, 0x10,000,000, where p is the price per product and x represents the number of items sold. The revenue for selling x products R=xp=x(500-.00005x). The total cost of producing x items is $50 per item plus a development cost of $2,500,000, so the total cost is C=50x+2500000. What price should the company charge per item to obtain a profit of at least $960,000,000?

We need to find both Revenue and Cost.

We need to know when this is greater than 960,000,000. We can do this graphically.

x=3500000, 5500000

Now substitute these solutions back into our equation for price.

The manufacturer will yield a profit of at least $960,000,000 if the price is between $225 and $325.

Recall, Domain is the inputs or possible x-values.

EX: Find the domain of .

We can confirm this graphically.

Upon completion of this lesson, you should be able to:

  1. Solve non-linear inequalities.
  2. Graphically
  3. Algebraically

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HW Pg. 204 6-66 6ths, 67, 72, 85-88

Quiz 2.5-2.7 tomorrow