1. Warm Up EQ: How do I perform operations on Rational Functions?
Add or subtract. 7 15 2 5 + 13 15 1. 11 12 3 8 – 2. 13 24
Simplify. Identify any x-values for which the expression is undefined.
4x9
12x3 3. 1 3 x6
x ≠ 0 1
x + 1
x – 1
x2 – 1 4.
x ≠ –1, x ≠ 1

2. EQ: How do I perform operations on Rational Functions?
Adding and subtracting rational expressions is similar to adding and subtracting fractions. To add or subtract rational expressions with like denominators, add or subtract the numerators and use the same denominator.

3. EQ: How do I perform operations on Rational Functions?
Example 1A: Adding and Subtracting Rational Expressions with Like Denominators
Add or subtract. Identify any x-values for which the expression is undefined.
x – 3
x + 4 +
x – 2
The expression is undefined at x = –4 because this value makes x + 4 equal 0.

4. EQ: How do I perform operations on Rational Functions?
Example 1B: Adding and Subtracting Rational Expressions with Like Denominators
Add or subtract. Identify any x-values for which the expression is undefined.
3x – 4
x2 + 1 –
6x + 1
There is no real value of x for which x2 + 1 = 0; the expression is always defined.

5. EQ: How do I perform operations on Rational Functions?
Check It Out! Example 1a
Add or subtract. Identify any x-values for which the expression is undefined.
6x + 5
x2 – 3 +
3x – 1
The expression is undefined at x = ± because this value makes x2 – 3 equal 0.

6. EQ: How do I perform operations on Rational Functions?
Check It Out! Example 1b
Add or subtract. Identify any x-values for which the expression is undefined.
3x2 – 5
3x – 1 –
2x2 – 3x – 2
Distribute the negative sign.
The expression is undefined at x = because this value makes 3x – 1 equal 0. 1 3

7. EQ: How do I perform operations on Rational Functions?
To add rational expressions with unlike denominators, rewrite both expressions with the LCD. This process is similar to adding fractions.

8. Example 3A: Adding Rational Expressions
EQ: How do I perform operations on Rational Functions?
Example 3A: Adding Rational Expressions
Add. Identify any x-values for which the expression is undefined.
x – 3
x2 + 3x – 4 + 2x
x + 4

9. EQ: How do I perform operations on Rational Functions?
Example 3A Continued
Add. Identify any x-values for which the expression is undefined.
x – 3 + 2x(x – 1)
(x + 4)(x – 1)
Add the numerators.
2x2 – x – 3
(x + 4)(x – 1)
Simplify the numerator.
Write the sum in factored or expanded form.
2x2 – x – 3
(x + 4)(x – 1)
x2 + 3x – 4 or
The expression is undefined at x = –4 and x = 1 because these values of x make the factors (x + 4) and (x – 1) equal 0.

10. Example 3B: Adding Rational Expressions
EQ: How do I perform operations on Rational Functions?
Example 3B: Adding Rational Expressions
Add. Identify any x-values for which the expression is undefined. x
x + 2 + –8
x2 – 4

11. EQ: How do I perform operations on Rational Functions?
Example 3B Continued
Add. Identify any x-values for which the expression is undefined.
Write the numerator in standard form.
x2 – 2x – 8
(x + 2)(x – 2)
(x + 2)(x – 4)
(x + 2)(x – 2)
Factor the numerator.
Divide out common factors.
x – 4
x – 2
The expression is undefined at x = –2 and x = 2 because these values of x make the factors (x + 2) and (x – 2) equal 0.

12. EQ: How do I perform operations on Rational Functions?
Check It Out! Example 3a
Add. Identify any x-values for which the expression is undefined. 3x
2x – 2 +
3x – 2
3x – 3

13. Check It Out! Example 3a Continued
EQ: How do I perform operations on Rational Functions?
Check It Out! Example 3a Continued
Add. Identify any x-values for which the expression is undefined.
9x + 6x – 4
6(x – 1)
Add the numerators.
15x – 4
6(x – 1)
Simplify the numerator.
The expression is undefined at x = 1 because this value of x make the factor (x – 1) equal 0.

14. EQ: How do I perform operations on Rational Functions?
Check It Out! Example 3b
Add. Identify any x-values for which the expression is undefined.
2x + 6
x2 + 6x + 9 + x
x + 3

15. Check It Out! Example 3b Continued
EQ: How do I perform operations on Rational Functions?
Check It Out! Example 3b Continued
Add. Identify any x-values for which the expression is undefined.
x2 + 5x + 6
(x + 3)(x + 3)
Write the numerator in standard form.
(x + 3)(x + 2)
(x + 3)(x + 3)
Factor the numerator.
Divide out common factors.
x + 2
x + 3
The expression is undefined at x = –3 because this value of x make the factors (x + 3) and (x + 3) equal 0.

16. Example 4: Subtracting Rational Expressions
EQ: How do I perform operations on Rational Functions?
Example 4: Subtracting Rational Expressions
Subtract Identify any x-values for which the expression is undefined.
2x2 – 30
x2 – 9 –
x + 3
x + 5

17. EQ: How do I perform operations on Rational Functions?
Example 4 Continued
Subtract Identify any x-values for which the expression is undefined.
2x2 – 30
x2 – 9 –
x + 3
x + 5
2x2 – 30 – x2 – 2x + 15
(x – 3)(x + 3)
Distribute the negative sign.
x2 – 2x – 15
(x – 3)(x + 3)
Write the numerator in standard form.
(x + 3)(x – 5)
(x – 3)(x + 3)
Factor the numerator.
x – 5
x – 3
Divide out common factors.
The expression is undefined at x = 3 and x = –3 because these values of x make the factors (x + 3) and (x – 3) equal 0.

18. EQ: How do I perform operations on Rational Functions?
Check It Out! Example 4a
Subtract Identify any x-values for which the expression is undefined.
3x – 2
2x + 5 –
5x – 2 2

19. Check It Out! Example 4a Continued
EQ: How do I perform operations on Rational Functions?
Check It Out! Example 4a Continued
Subtract Identify any x-values for which the expression is undefined.
3x – 2
2x + 5 –
5x – 2 2
15x2 – 16x + 4 – 4x – 10
(2x + 5)(5x – 2)
Distribute the negative sign. 5 2 2 5
The expression is undefined at x = – and x = because these values of x make the factors (2x + 5) and (5x – 2) equal 0.

20. EQ: How do I perform operations on Rational Functions?
Check It Out! Example 4b
Subtract Identify any x-values for which the expression is undefined.
2x2 + 64
x2 – 64 –
x + 8
x – 4

21. EQ: How do I perform operations on Rational Functions?
Check It Out! Example 4b
Subtract Identify any x-values for which the expression is undefined.
2x2 + 64
x2 – 64 –
x + 8
x – 4
2x – x2 + 12x – 32)
(x – 8)(x + 8)
Distribute the negative sign.
x2 + 12x + 32
(x – 8)(x + 8)
Write the numerator in standard form.
(x + 8)(x + 4)
(x – 8)(x + 8)
Factor the numerator.
x + 4
x – 8
Divide out common factors.
The expression is undefined at x = 8 and x = –8 because these values of x make the factors (x + 8) and (x – 8) equal 0.

22. EQ: How do I perform operations on Rational Functions?
Some rational expressions are complex fractions. A complex fraction contains one or more fractions in its numerator, its denominator, or both. Examples of complex fractions are shown below.
Recall that the bar in a fraction represents division. Therefore, you can rewrite a complex fraction as a division problem and then simplify. You can also simplify complex fractions by using the LCD of the fractions in the numerator and denominator.

23. Example 5A: Simplifying Complex Fractions
Simplify. Assume that all expressions are defined.
x + 2
x – 1
x – 3
x + 5
Write the complex fraction as division.
x + 2
x – 1
x – 3
x + 5 ÷
Write as division.
x + 2
x – 1
x + 5
x – 3 
Multiply by the reciprocal.
x2 + 7x + 10
x2 – 4x + 3
(x + 2)(x + 5)
(x – 1)(x – 3) or
Multiply.

24. Example 5B: Simplifying Complex Fractions
Simplify. Assume that all expressions are defined.
x – 1 x 2 3 +
Multiply the numerator and denominator of the complex fraction by the LCD of the fractions in the numerator and denominator.
x – 1 x
(2x) 2 3 +
The LCD is 2x.

25. Example 5B Continued
Simplify. Assume that all expressions are defined.
(3)(2) + (x)(x)
(x – 1)(2)
Divide out common factors.
x2 + 6
2(x – 1)
x2 + 6
2x – 2 or
Simplify.

26. Simplify. Assume that all expressions are defined.
Check It Out! Example 5a
Simplify. Assume that all expressions are defined.
x + 1
x2 – 1 x
x – 1
Write the complex fraction as division.
x + 1
x2 – 1 x
x – 1 ÷
Write as division.
x + 1
x2 – 1
x – 1 x 
Multiply by the reciprocal.

27. Check It Out! Example 5a Continued
Simplify. Assume that all expressions are defined.
x + 1
(x – 1)(x + 1)
x – 1 x 
Factor the denominator. 1 x
Divide out common factors.

28. Simplify. Assume that all expressions are defined.
Check It Out! Example 5b
Simplify. Assume that all expressions are defined. 20
x – 1 6
3x – 3
Write the complex fraction as division. 20
x – 1 6
3x – 3 ÷
Write as division. 20
x – 1
3x – 3 6 
Multiply by the reciprocal.

29. Check It Out! Example 5b Continued
Simplify. Assume that all expressions are defined. 20
x – 1
3(x – 1) 6 
Factor the numerator. 10
Divide out common factors.

30. Simplify. Assume that all expressions are defined.
Check It Out! Example 5c
Simplify. Assume that all expressions are defined.
x + 4
x – 2 1 2x x +
Multiply the numerator and denominator of the complex fraction by the LCD of the fractions in the numerator and denominator.
x + 4
x – 2
(2x)(x – 2) 1 2x x +
The LCD is (2x)(x – 2).

31. Check It Out! Example 5c Continued
Simplify. Assume that all expressions are defined.
(2)(x – 2) + (x – 2)
(x + 4)(2x)
Divide out common factors.
3x – 6
(x + 4)(2x)
3(x – 2)
2x(x + 4) or
Simplify.

32. Example 6: Transportation Application
A hiker averages 1.4 mi/h when walking downhill on a mountain trail and 0.8 mi/h on the return trip when walking uphill. What is the hiker’s average speed for the entire trip? Round to the nearest tenth.
Total distance: 2d
Let d represent the one-way distance. d
0.8
1.4 +
Total time:
Use the formula t = r d
The average speed is
total time
total distance d
0.8
1.4 + 2d
Average speed:

33. Example 6 Continued d
0.8
1.4 + 2d
= = and = =
1.4 d 5 7 5d
0.8 4 5d 4 7 +
2d(28)
(28)
The LCD of the fractions in the denominator is 28.
56d
20d + 35d
Simplify.
55d
≈ 1.0
Combine like terms and divide out common factors.
The hiker’s average speed is 1.0 mi/h.

34. Check It Out! Example 6
Justin’s average speed on his way to school is 40 mi/h, and his average speed on the way home is 45 mi/h. What is Justin’s average speed for the entire trip? Round to the nearest tenth.
Total distance: 2d
Let d represent the one-way distance. d 45 40 +
Total time:
Use the formula t = r d
The average speed is
total time
total distance d 45 40 + 2d
Average speed:

35. Check It Out! Example 6 d 40
(360) 45 +
2d(360)
The LCD of the fractions in the denominator is 360.
720d
9d + 8d
Simplify.
Combine like terms and divide out common factors.
720d
17d
≈ 42.4
Justin’s average speed is 42.4 mi/h.

36. Lesson Quiz: Part I
Add or subtract. Identify any x-values for which the expression is undefined.
3x2 – 2x + 7
(x – 2)(x + 1)
x ≠ –1, 2
2x + 1
x – 2 +
x – 3
x + 1 1. x
x + 4 –
x + 36
x2 – 16
x – 9
x – 4 2.
x ≠ 4, –4
3. Find the least common multiple of x2 – 6x and x2 + x – 2.
(x – 5)(x – 1)(x + 2)

37. 4. Simplify . Assume that all expressions are defined. 1 x
Lesson Quiz: Part II
x + 2
x2 – 4 x
x – 2
4. Simplify Assume that all expressions are defined. 1 x
5. Tyra averages 40 mi/h driving to the airport during rush hour and 60 mi/h on the return trip late at night. What is Tyra’s average speed for the entire trip?
48 mi/h