This document discusses solving rational equations and functions. It provides examples of solving rational equations by multiplying both sides by the LCD, checking for extraneous solutions, and solving word problems involving rates and distances using rational functions. It also gives an example of solving a word problem about a round trip flight with headwinds and tailwinds to determine the wind speed.

1. Unit 4 Rational functions
8-6 Solving Rational Expressions

2. 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)

3. SOLVE: LCD?
( x + 3)( x − 3)
 x x  2 
( x + 3)( x − 3) 
 x − 3 + x + 3 ( xx+ 3)(9 − 3)  ( x + 3)( x − 3)
=
 2 − x 
  
Multiply by LCD
SOLVE x( x + 3) + x( x − 3) = 2
x 2 + 3x + x 2 − 3x = 2
Check for Extraneous solutions.
2x = 2
2
(plug answer into original problem
1 1 2 1 1 2
x =1
2
+ = + =
1 − 3 1 + 3 12 − 9 − 1 − 3 − 1 + 3 (−1) − 9
2
x = ±1 1 1 2 1 1 2
+ = + =
−2 4 −8 −4 −82
1 1 1 1
− =− − =−
4 4 4 4

4. SOLVE: LCD?
( x + 2)( x + 1)
 x −1 2 x   x − 1 ( x + 2)( x + 1)

( x + 2)( x + 1)  2
 +  =
  

( xx+ 2)( x + 1) x + 2 x + 1
 +3 2  
x − 1 + 2 x( x + 1) = ( x − 1)( x + 2)
x −1+ 2x + 2x = x + x − 2
2 2
2 x + 3x − 1 = x + x − 2
2 2
x2 + 2x +1 = 0
CHECK ??
( x + 1)( x + 1) = 0 -1 does not work so
there is NO SOLUTION
x = −1

5. 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).
2 x −2 2 x+2
+ =1 = +
x+2 x−2 x + x x x +1
2
Answer: X=0 Answer: X= -2

6. Practice
• Pg 546 #14-19, 44-52(even)

7. 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.
100(0.02) + x(0.005)
y=
100 + x
• How man ML of the 0.5% solution must be
added to have a combined concentration of
0.9%?
100(0.02) + x(0.005)
.009 =
100 + x
Answer: (graph? solve X=275
by hand)

8. • 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. 75 + 100 x
Answer: f ( x) =
Find when the rational x +1
function will be 95%
75 + 100 x
95 =
x +1
To find the average of Answer: (graph) You will need to make
something: add and divide 100% on the next 4 test to
by total number bring your test average
up to a 95%

9. Distance = rate x time
When a problem involves “how fast”, “how far”, or “for how long”, think about distance
equation d =rt
• A plane flies 1850 miles at a speed of 480 mph. Find the time of the trip to
the nearest hundredth. 1850 = 480(t )
Answer: t = 3.85
• 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.
1850 = (480 + x)(3.7)
Answer: x = 20

10. 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?
Working together Mary and Sally
How much of the job can Joe do in can paint a fence in 5.14 hours.
one hour? Steve? Working alone Mary can paint the
fence in 9 hours. How long would it
1 1 1
+ = take Sally to paint the fence alone?
8 12 t 1 1 1
+ =
5 1 24 9 x 5.14
= =t
24 t 5 x = 11.98

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

12. A plane leaves Chicago and flies to San Rate * time = distance
Francisco (1850 miles away) with a headwind r *t = d Time of a trip equals
the distance over
The same plane returns to Chicago with a speed d
tailwind. The round trip took 7.75 hours.If the 1850 t=
t= r
airplane cruises at 480 mph, what is the speed 480 − x
of the wind? (Assume the winds are constant.)
X=wind speed
San Francisco
Chicago
1850 1850
7.75 = +
480 + x 480 − x
1850
t=
Solve by Graphing 480 + x
Zoom fit, x-max to 50
X = 35.195 Write an equation