Do Now: Solve for x:

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Aim: HOW DO WE SOLVE INEQUALITIES? HW: p.A73 #11-24 even

http://my.hrw.com/math06_07/nsmedia/tools/Graph_Calculator/graphCalc.html

5 7 3 11x x Do Now: Solve for x:

Using interval notation:

Solution:

9,

or 9x 9 x

WHEN MULTIPLYING OR DIVIDING BY A NEGATIVE, REMEMBER TO REVERSE THE INEQUALITY SIGN!!!

•Parenthesis does not include the endpoint.

•Bracket includes the endpoint.

•Infinity never gets a bracket.

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Solve for x and show your answer in interval notation and on a number line:

3 4 23x

5x

5,

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Consider two cases:

II. Quadratic < 0I. Quadratic > 0

0x a x b

0 0x a x b

0 0x a x b

0 0x a x b

0 0x a x b

0x a x b

Solving Quadratic Inequalities algebraically:

HW p. A73 # 55,56

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Solve the Quadratic Inequality algebraically:

3 2 0x x

3 0 2 0x x

3 2x x 3 0 2 0x x

3 2x x

2,32 3x

HW p. A73 # 55,56

2 6 0x x

3 0 2 0 3 0 2 0x x x x

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Solving Quadratic Inequality graphically:2 2 3 7x x

4.3166,2.3166

21

2

2 3

7

y x x

y

Since we want the inequality to work for all values of

that are below 7, we want all the values between the left and right POI’s (See shaded region). Thus the solution is:

2 2 3 7x x

HW p. A73 # 55,56

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Quadratic Inequality Application:

A projectile is fired straight upward from ground level with an initial velocity of 384 feet per second. During what time period will its height exceed 2000 feet?

20 016s t v t s

The position of an object moving vertically can be modeled by the position equation:

s is the height and t is the time. In this case s0 = 0, v0 = 384, and s must be greater than 2000 so:

216 384 2000t t

HW p. A74 #81, 83

216 384 0s t t

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216 384 2000t t You must adjust the Zoom and Window settings to get a clear picture of the intersection of the 2 graphs.

Try Zoom Fit and change the Window settings as indicated in the image:

7.64,16.36

HW p. A74 #81, 83

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HW p. A74 #81, 83

SOLUTIONS TO HOMEWORK PROBLEMS (Quad Ineq. Apps)

#81: A projectile is fired straight upward from ground level with an initial velocity of 160 feet per second.

(a) At what instant will it be back at ground level?

(b) When will the height exceed 384 feet?

216 160s t t

216 160 384t t

(a)

(b)

20 16 160t t When the projectile is at ground level the height, s, is zero. We are finding t when s = 0.

100,t t

21 16 160y t t 2 384y

Range of t values when the height of the projectile is above 384

4,6

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SOLUTIONS TO HOMEWORK PROBLEMS (Quad Ineq. Apps)

#83: The numbers D of doctorate degrees (in thousands) awarded to female students from 1990 to 2003 in the US can be approximated by the following model, where t is the year, with t = 0 corresponding to 1990:

HW p. A74 #81, 83

(a) Use a graphing utility to graph the model

20.0165 0.755 14.06D t t 0 13t

21 0.0165 0.755 14.06y t t

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(b) Use the zoom and trace features to find when the number of degrees was between 15 and 20 thousand.

21 0.0165 0.755 14.06y t t

2 15y

3 20y

21 0.0165 0.755 14.06y t t 2 15y 3 20y

21 0.0165 0.755 14.06y t t

2 15y

3 20y

The blue line indicates the interval where the curve is between the 15 and 20

1.28 10.09t

1991,2000

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(d) According to the model, will the number of degrees exceed 30 thousand? If so, when? If not, explain.

(c) Algebraically verify your results from part (b)

215 0.0165 0.755 14.06 20t t 15 20D

1.2899,44.1646

2

2

0.0165 0.755 14.06 15

0.0165 0.755 0.94 0

t t

t t

2

2

0.0165 0.755 14.06 20

0.0165 0.755 5.94 0

t t

t t

,10.0945 35.6631,

Show using a number line why the solution is:

1.28 10.09t

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2.51 179.6,0 5

3.37 57.9,0 5

N t t

V t t

Use the models which approximate the annual numbers of hours per person spent reading daily newspapers N and playing video games V for the years 2000 to 2005, where t is the year, with t = 0 corresponding to 2000.

( ) 65V t ( ) 175N t ( ) ( )V t N t( ) ( )V t N t

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Review the 4 situations when dealing with conjunctions and disjunctions:

2 6 0x x

2 2 8 0x x

2 2 3 0x x

2 20 0x x

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Do Now: Solve for x: