Unit 4 Rational functions8-6 Solving rational expressions

Solving Rational Equations•To solve a rational equation first multiply

each side by the LCD.

•You must check for extraneous solutions. (A solution that doesn’t work)

SOLVE:

9

2

33 2

xx

x

x

x

LCD?

)3)(3( xx

)3)(3( xx

)3)(3( xx )3)(3( xx

)3( xx )3( xx 2

233 22 xxxx

22 2 x

12 x1x

Multiply by LCD

SOLVE

Check for Extraneous solutions. (plug answer into original problem

91

2

31

1

31

12

8

2

4

1

2

1

4

1

4

1

9)1(

2

31

1

31

12

8

2

2

1

4

1

4

1

4

1

SOLVE:

1

1

2

2

23

12

x

x

x

x

xx

x

)1)(2( xx

LCD?

)1)(2( xx

)1)(2( xx

)1)(2( xx

1x )1(2 xx )2)(1( xx

xxx 221 2

132 2 xx

22 xx

22 xx

0122 xx0)1)(1( xx

1x

CHECK ??-1 does not work so there is NO SOLUTION

How to solve using the CALC.• Set equal to zero and graph (have to use parenthesis.• Y2 = 0• Find where the graph crosses x-axis (2nd trace intersect).

122

2

x

x

x 1

2222

x

x

xxx

Answer: Answer:X=0 X= -2

9

2

33 2

xx

x

x

x

Answer: X= 1, -1

Practice

•Pg 546 #8-19(all), 44-52(even)

Word Problems•The function below gives the concentration

of the saline solution after adding x milliliters of 0.5% solution to 100 milliliters of 2% solution.

•How man ML of the 0.5% solution must be added to have a combined concentration of 0.9%?

Answer: (graph? solve by hand)

X=275

x

xy

100

)005.0()02.0(100

x

x

100

)005.0()02.0(100009.

•You earn a 75% on the first test of the quarter how many consecutive 100% test scores do you need to bring your test average up to a 95%?

Write a rational function.

Find when the rational function will be 95%

Answer:

1

10075)(

x

xxf

You will need to make 100% on the next 4 test to bring your test average up to a 95%

1

1007595

x

x

To find the average of something: add and divide by total number

Answer: (graph)

Distance = rate x time

• A plane flies 1850 miles at a speed of 480 mph. Find the time of the trip to the nearest hundredth.

• On the return trip the plane travels at the same speed (480) but a tail wind helps the plane move faster. The total flying time for the round trip is 7.55 hours. Find the speed x of the tail wind.

85.3t

20x

Answer:

Answer:

)(4801850 t

)7.3)(480(1850 x

When a problem involves “how fast”, “how far”, or “for how long”, think about distance equation d =rt

Work Problems• “you have to think of the problem in terms of how much

each person or thing does in a given amount of time”

Joe can paint a room in 8 hours and Steve can paint the same room in 12 hours. How long will it take the two to paint the room if they work together?

How much of the job can Joe do in one hour? Steve?

8

1

12

1

t

1

t

1

24

5 t

5

24

Working together Mary and Sally can paint a fence in 5.14 hours. Working alone Mary can paint the fence in 9 hours. How long would it take Sally to paint the fence alone?

9

1

x

1

14.5

1

98.11x

Practice

•Pg 521 #35, 39, 40, 46*•Pg 545 #20,36,37,38,39

A plane leaves Chicago and flies to San Francisco (1850 miles away) with a headwind

The same plane returns to Chicago with a tailwind. The round trip took 7.75 hours.If the airplane cruises at 480 mph, what is the speed of the wind? (Assume the winds are constant.)

ChicagoSan Francisco

Rate * time = distance

r

dt

dtr *

xt

480

1850

xt

480

1850xx

480

1850

480

185075.7

X=wind speed

Solve by Graphing

Zoom fit, x-max to 50

X = 35.195

Time of a trip equals the distance over speed

Write an equation