8-3
Adding and Subtracting
Rational Expressions
Some rational expressions are complex fractions.
A complex fraction contains one or more
fractions in its numerator, its denominator, or
both. Examples:
The bar in the fraction represents division.
Therefore, you can rewrite a complex fraction as a
division problem.
Holt Algebra 2
8-3
Adding and Subtracting
Rational Expressions
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–3
Write as division.
x–1
x+5
Multiply by the
x+2
x+5

reciprocal.
x–1
x–3
(x + 2)(x + 5) or x2 + 7x + 10
(x – 1)(x – 3)
x2 – 4x + 3
Holt Algebra 2
Multiply.
Adding and Subtracting
Rational Expressions
8-3
Example 5B: Simplifying Complex Fractions
Simplify. Assume that all expressions are defined.
3
x
+ 2
x
x–1
x
Add the rational functions in the numerator.
(2)
(2)
3
x (x)
+
x
2 (2)
x–1
x
Holt Algebra 2
The LCD is 2x.
8-3
Adding and Subtracting
Rational Expressions
Example 5B Continued
Simplify. Assume that all expressions are defined.
x2 + 6
2x

x
x–1
x3 + 6x
2x2 – 2x
x(x2 + 6)
2x(x – 1)
x2 + 6
2(x – 1)
Holt Algebra 2
Rewrite as multiplication of
the reciprocal.
Simplify.
8-3
Adding and Subtracting
Rational Expressions
Check It Out! Example 5b
Simplify. Assume that all expressions are defined.
20
x–1
3x – 3
Write the complex fraction as division.
20 ÷ 3x – 3
x–1
1
Write as division.
20 
1
x–1
3x – 3
Multiply by the reciprocal.
Holt Algebra 2
8-3
Adding and Subtracting
Rational Expressions
Check It Out! Example 5b Continued
Simplify. Assume that all expressions are defined.
20
1

x–1
3x – 3
20
(x – 1)(3x – 3)
20
3x2 – 6x + 3
Holt Algebra 2
Multiply
8-3
Adding and Subtracting
Rational Expressions
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.
Total time: d + d
1.4 0.8
Use the formula t = d
r .
Average speed:
Holt Algebra 2
2d
d + d
1.4 0.8
The average speed is
total distance .
total time
8-3
Adding and Subtracting
Rational Expressions
Example 6 Continued
2d
d = d = 5d and d = d = 5d .
1.4
7
7
0.8
4
4
5
5
2d(28)
The LCD of the fractions in the
5d (28) + 5d (28)
denominator is 28.
7
4
56d
Simplify.
20d + 35d
d + d
1.4 0.8
55d ≈ 1.0
55d
Combine like terms and
divide out common factors.
The hiker’s average speed is 1.0 mi/h.
Holt Algebra 2
8-3
Adding and Subtracting
Rational Expressions
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.
Total time: d + d
40
45
Use the formula t = d
r .
Average speed:
Holt Algebra 2
2d
d + d
40
45
The average speed is
total distance .
total time
8-3
Adding and Subtracting
Rational Expressions
Check It Out! Example 6
2d(360)
d (360)+ d (360)
40
45
720d
9d + 8d
720d ≈ 42.4
17d
The LCD of the fractions in the
denominator is 360.
Simplify.
Combine like terms and
divide out common factors.
Justin’s average speed is 42.4 mi/h.
Holt Algebra 2
8-3
Adding and Subtracting
Rational Expressions
HW pg. 588
#’s 28-30, 43-45
Holt Algebra 2