1. Add and Subtract Rational Expressions
Warm Up
Lesson Presentation
Lesson Quiz

2. Warm-Up
1. Find 1 4 2 3
ANSWER 11 12
2. Find – 7 8 5 6
ANSWER 1 24 3.
A boat travels 8 miles upstream (against the current) in 4 hours, and 8 miles downstream (with the current) in 3.2 hours. What is the average speed of the boat in still water?
ANSWER
2.25 mi/h

3. Example 1 a. 5 3x 7 + = 12 3x 3 4 3 x = = 4 x b. 3x x – 1 – = x + 5
Add numerators.
3 4
3 x =
Factor and divide out common factor. = 4 x
Simplify. b. 3x
x – 1 – =
x + 5
3x – (x + 5)
x – 1
Subtract numerators.
2x – 5
x – 1 =
Simplify.

4. Example 1
CHECK
Check your simplification using a graphing calculator. For part (b), graph 3x
x – 1 –
y1=
x + 5
and
2x – 5
y2=
The graphs coincide. So, the expressions are equivalent for all values of x other than the excluded value of 1.

5. Guided Practice
Find the sum or difference. 1. 2 y +
y + 1 =
y + 3 y 2.
4x + 1
2x – 1
2x – 3 –
2x + 4
2x – 1 =

6. Find the LCD of the rational expressions. r + 3 10r2 a. , 1 4r 5
Example 2
Find the LCD of the rational expressions.
r + 3
10r2
a , 1 4r 5
(x – 3)2
b ,
3x + 4
x2 – x – 6
c , 3
c – 2
c + 8
2c + 7
SOLUTION
a. Find the least common multiple (LCM) of 4r and 10r2.
4r = r
The common factors are circled.
10r2 = r r
LCM = 2 r r = 20r2
ANSWER
The LCD of and is 20r2. 1 4r
r + 3
10r2

7. Example 2 5
(x – 3)2
b ,
3x + 4
x2 – x – 6
SOLUTION
b. Find the least common multiple (LCM) of (x – 3)2
and x2 – x – 6.
(x – 3)2 = (x – 3) (x – 3)
x2 – x – 6 = (x – 3) (x + 2)
LCM = (x – 3) (x – 3) (x + 2) = (x – 3)2(x + 2)
ANSWER
The LCD of and is (x – 3)2(x + 2). 5
(x – 3)2
3x + 4
x2 – x – 6

8. Example 2
c , 3
c – 2
c + 8
2c + 7
SOLUTION
c. Find the least common multiple of c – 2 and 2c + 7.
Because c – 2 and 2c + 7 cannot be factored, they don’t have any factors in common. The least common multiple is their product, (c – 2)(2c + 7).
ANSWER
The LCD of and is (c – 2)(2c + 7). 3
c – 2
c + 8
2c + 7

9. Guided Practice
Find the LCD of the rational expressions.
m + 1
7m3 , 1
28m
ANSWER
The LCD of and is 28m3. 1
28m
m + 1
7m3 ,
x2 + 2
x2 + 7x + 10 2
x2 + 4x – 5
ANSWER
The LCD of and is (x + 5)(x – 1)(x + 2). 2
x2 + 4x – 5
x2 + 2
x2 + 7x + 10

10. Guided Practice
Find the LCD of the rational expressions. , 5a
a + 3
a + 6
a – 4
ANSWER
The LCD of and is (a + 3)(a – 4). 5a
a + 3
a + 6
a – 4

11. Example 3 Find the sum . 5 12x3 9 8x2 + 5 12x3 9 8x2 + = 9 3x 8x2 3x
5 2
12x3 2
Rewrite fractions using LCD, 24x3.
27x
24x3 = + 10
Simplify numerators and denominators. =
27x + 10
24x3
Add fractions.

12. Example 4 Find the difference – . 7x x + 2 10 3x 7x x + 2 10 3x = –
Rewrite fractions using
LCD, 3x(x + 2).
10(x + 2) – 7x(3x)
3x(x + 2) =
Subtract fractions.
– 21x2 + 10x + 20
3x(x + 2) =
Simplify numerator.

13. Example 5 Find the difference . x + 4 x2 + 3x – 10 – x – 1 x2 + 2x – 8=
x + 4
(x – 2)(x + 5) –
x – 1
(x + 4)(x – 2)
Factor denominators. =
(x + 4)(x + 4)
(x – 2)(x + 5) (x + 4) –
(x – 1)(x + 5)
(x + 4)(x – 2)(x + 5)
Rewrite fractions using
LCD, (x – 2)(x + 5)(x + 4). =
(x + 4)(x + 4) – (x – 1)(x + 5)
(x – 2)(x + 5)(x + 4)
Subtract fractions.

14. Example 5 x2 + 8x + 16 – (x2 + 4x – 5) = (x – 2)(x + 5)(x + 4) 4x + 21
Find products in numerator. =
4x + 21
(x – 2)(x + 5)(x + 4)
Simplify.

15. Guided Practice
Find the sum or difference. 3 2x 7
5x 4 =
15x3 + 14
10x 4 6. + 7. y
y + 1 + 3
y + 2 =
y2 + 5y + 3
( y +1)( y + 2) 8.
2z – 1
z2 + 2z – 8 –
z + 1
z2 – 4 =
z2 – 2z – 6
(z + 4)(z – 2)(z + 2)

16. Example 6
BOAT TRAVEL
A boat travels 24 kilometers upstream (against the current) and 24 kilometers downstream (with the current) as shown in the diagram. Write an equation that gives the total travel time t (in hours) as a function of the boat’s average speed r (in kilometers per hour) in still water. Find the total travel time if the boat’s average speed in still water is 10 kilometers per hour.

17. Example 6
SOLUTION
STEP 1
Write a verbal model. Then write an equation.

18. Find the sum of the expressions on the right side of the equation.
Example 6
SOLUTION
STEP 2
Find the sum of the expressions on the right side of the equation.
t = 24
r – 2 +
r + 2
Write equation. =
24(r + 2)
(r – 2) (r + 2) +
24(r – 2)
(r + 2) (r – 2)
Rewrite fractions using LCD, (r – 2)(r + 2). =
24(r + 2) + 24(r – 2)
(r – 2) (r + 2)
Add fractions. =
48r
(r – 2) (r + 2)
Simplify.

19. Example 6
STEP 3
Calculate the value of t when r = 10.
t =
48(10)
(10 – 2)(10 + 2) =
480
(8)(12) 96
= 5
ANSWER
The total travel time is 5 hours.

20. Guided Practice 9.
WHAT IF? In Example 6, suppose the speed of the current is 3 kilometers per hour. Find the total travel time.
ANSWER
The total travel time is about 5.3 hours.

21. Lesson Quiz
Find the sum or difference. 1. 4 2x +
x +1 3
ANSWER
x2 + x + 6 3x 2.
x – 6
x – 1 3x –
ANSWER
2x2 – 16x – 1
(x – 1)(3x) 3. 2x
x2 + 6x – 16
x + 3
x2 – 3x + 2 +
ANSWER
3x2 + 9x + 24
(x + 8 ) (x – 2) (x – 1)

22. Lesson Quiz
You hike 4 miles up a hill and then 4 miles back down the hill. Your speed hiking up the hill is 1.5 miles per hour less than your speed hiking back down. Find the total time of your hike if you hiked 2.5 miles per hour up the hill. 4.
ANSWER
5.6 h