copyright © 2011 pearson education, inc. slide 4.3-1 4.3 rational equations, inequalities,...

Copyright © 2011 Pearson Education, Inc. Slide 4.3-1

4.3 Rational Equations, Inequalities, Applications, and Models

• Solving Rational Equations and Inequalities– at least one variable in the denominator

– may be undefined for certain values where the denominator is 0

– identify those values that make the equation (or inequality) undefined

– when solving rational equations, you generally multiply both sides by a common denominator

– when solving rational inequalities, you generally get 0 on one side, then rewrite the rational expression as a single fraction


4.3 Solving a Rational Equation Analytically

Example Solve

Analytic Solution Notice that the expression is

undefined for

.1122

xx

.21x

1122

xx

Multiply both sides by 2x + 1.)12(12 xx

xxx

1122

Solve for x.

The solution set is {1}.


4.3 Solving a Rational Equation Analytically

Graphical Solution Rewrite the equation as

and define Y1 Using the

x-intercept method shows that the zero of the function

is 1.

01122

xx

.1122

xx


4.3 Solving a Rational Equation

Example Solve

Solution For this equation,

.4

82

12 2

xxxx

.2x

48

21

2 2

xxxx

8)2(1)2( xxx

8222 xxx

01032 xx


4.3 Solving a Rational Equation

But, x = 2 is not in the domain of the original

equation and, therefore, must be rejected. The

solution set is {–5}.

2or5

0)2)(5(

xx

xx


4.3 Solving a Rational Inequality Analytically

Example Solve the rational inequality

Analytic Solution We can’t multiply both sides by 2x + 1 since it may be negative. Start by subtracting 1 from both sides.

.1122

xx

01212

122

01122

xx

xx

xx

Common denominator is 2x + 1.



To determine the sign graph, solve the equations

to get x = 1 and

0121

012

122

012

)12(2

xx

xxx

xxx

Rewrite as a single fraction.

012and01 xx

.21x



Complete the sign graph and determine the intervalswhere the quotient is negative.

The quotient is zero or negative when x is in Can’t include it makes the denominator 0.

).,1[),(21

;21


4.3 Solving a Rational Inequality Graphically

Graphical Solution Let Y1 We use the

graph to find the intervals where Y1 is below the

x-axis, including the x-intercepts, where Y1 = 0.

The solution set is

.1122

xx

).,1[),(21


4.3 Solving Equations Involving Rational Functions

Solving a Rational Equation

1. Rewrite the inequality, if necessary, so that 0 is on one side and there is a single rational expression on the other side.

2. Determine the values that will cause either the numerator or the denominator of the rational expression to equal 0. These values determine the intervals on the number line to consider.

3. Use the test value from each interval to determine which intervals form the solution set. Be sure to check endpoints.


4.3 Models and Applications of Rational Functions: Analyzing Traffic Intensity

Example Vehicles arrive randomly at a parking ramp at an

average rate of 2.6 vehicles per minute. The parking attendant can admit 3.2 cars per minute. However, since arrivals are random, lines form at various times.

(a) The traffic intensity x is defined as the ratio of the average arrival rate to the average admittance rate. Determine x for this parking ramp.

(a) The average number of vehicles waiting in line to enter the ramp is modeled by f (x) = where 0 x <1 is the traffic intensity. Compute f (x) for this parking ramp.

(c) Graph y = f (x). What happens to the number of vehicles waiting as the traffic intensity approaches 1?

2

2(1 ) ,xx


4.3 Models and Applications of Rational Functions: Analyzing Traffic Intensity

Solution(a) Average arrival rate = 2.6 vehicles/min,

average admittance rate = 3.2 vehicles/min, so

(b) From part (a), the average number of vehicles waiting in line is f (.8125).

.8125.2.36.2 x

vehicles76.1)8125.1(2

8125.)8125(.

2

f


4.3 Models and Applications of Rational Functions: Analyzing Traffic

Intensity(c) From the graph below, we see that as x

approaches 1, y = f (x) gets very large, that is, the number of waiting vehicles gets very large.


4.3 Models and Applications of Rational Functions: Optimization Problem

Example A manufacturer wants to construct cylindrical aluminum cans with volume 2000 cm3 (2 liters). What radius and height will minimize the amount of aluminum used? What will this amount be?

Solution Two unknowns: radius xand height h. To minimize the amount of aluminum, we minimize the surface area. Volume V is

.2000

So.2000 22

2

xhhx

hxV


4.3 Models and Applications of Rational Functions: Optimization Problem

Surface area S = 2xh + 2x2, x > 0 (since x is the radius), can now be written as a function of x.

Minimum radius is approximately 6.83 cm and the height

associated with that is 13.65 cm, giving a minimum

amount of aluminum of 878.76 cm3.

xx

xx

xx

xxS

3

2

22

24000

24000

22000

2)(

2)83.6(2000


4.3 Inverse Variation

Inverse Variation as the nth PowerLet x and y denote two quantities and n be a positive number. Then y is inversely proportional to the nth power of x, or y varies inversely as the nth power of x, if there exists a nonzero number k such that

If then y is inversely proportional to x, or y

varies inversely as x.

.n

ky

x

,k

yx


4.3 Modeling the Intensity of Light

The intensity of light I is inversely proportional to the second power of the distance d. The equation

models this phenomenon. At a distance of 3 meters, a 100-watt bulb produces an intensity of 0.88 watt per square meter. Find the constant of variation k, and then determine the intensity of the light at a distance of 2 meters.

Substitute d = 3, and I = .88 into the variation equation, and solve for k.

2

kI

d

2

20.88

37.92

kI

dk

k

2

2

7.92

7.92

21.98

Id

I

I


4.3 Joint Variation

Joint VariationLet m and n be real numbers. Then z varies jointly as the nth power of x and the mth power of y if a nonzero real number k exists such that

z = kxnym.


4.3 Solving a Combined Variation Problem

In the photography formula

the luminance L (in foot-candles) varies directly as the square of the F-stop F and inversely as the product of the file ASA number s and the shutter speed t. The constant of variation is 25.Suppose we want to use 200 ASA file and a shutter speed of 1/250 when 500 foot candles of light are available. What would be an appropriate F-stop?

225FL

st

225500

1200

250

F

2400 25F

216 F4 F

An F-stop of 4 would be appropriate.


4.3 Rate of Work

Rate of WorkIf 1 task can be completed in x units of time, then the rate of work is 1/x task per time unit.


4.3 Analyzing Work Rate

Example It takes machine B one hour less to complete a task when working alone than it takes machine A working alone. If they start together, they can complete the task in 72 minutes. How long does it take each machine to complete the task when working alone?

Solution Let x represent the number of hours it takes machine A to complete the task alone. Then it takes machine B hours working alone.

.2000

So.2000 22

2

xhhx

hxV

1x


4.3 Analyzing Work Rate

Solution

The only value that makes sense is 3. It takes machine A 3 hours to complete the task alone, and it takes machine B 2 hours to complete the task alone.

2

2

1 1 5

1 61 1 5

6 1 6 11 6

6 1 6 5 16 6 6 5 5

0 5 17 60 5 2 3

x x

x x x xx xx x x xx x x x

x xx x

5 2 0 or 3 02

or 35

x x

x x

copyright © 2011 pearson education, inc. slide 4.3-1 4.3 rational equations, inequalities,...

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