Adding
and
Subtracting
Adding
and
Subtracting
5-3
5-3 Rational Expressions
Rational Expressions
Warm Up
Lesson Presentation
Lesson Quiz
HoltMcDougal
Algebra 2Algebra 2
Holt
5-3
Adding and Subtracting
Rational Expressions
Warm Up
Add or subtract.
2 + 7
1. 15
15
11 – 3
2. 12
8
Simplify.
3.
4.
4x9
12x3
x– 1
x2 – 1
Holt McDougal Algebra 2
9
15
13
24
1 x6
3
1
x+1
or
3
5
5-3
Adding and Subtracting
Rational Expressions
Objectives
Add and subtract rational expressions.
Simplify complex fractions.
Holt McDougal Algebra 2
5-3
Adding and Subtracting
Rational Expressions
Vocabulary
complex fraction
Holt McDougal Algebra 2
5-3
Adding and Subtracting
Rational Expressions
Essential Question
• How do you add or subtract rational
expressions with the same denominator?
Holt McDougal Algebra 2
5-3
Adding and Subtracting
Rational Expressions
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.
Holt McDougal Algebra 2
5-3
Adding and Subtracting
Rational Expressions
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–2
x+4
x+4
x–3+x–2
Add the numerators.
x+4
2x – 5
Combine like terms.
x+4
The expression is undefined at x = –4. To find this
value, set the denominator equal to 0 and solve.
This will make x + 4 equal 0 and x = −4.
Holt McDougal Algebra 2
5-3
Adding and Subtracting
Rational Expressions
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 – 6x + 1
x2 + 1
x2 + 1
3x – 4 – (6x + 1)
x2 + 1
3x – 4 – 6x – 1
x2 + 1
–3x – 5
x2 + 1
There is no real value of
the expression is always
Holt McDougal Algebra 2
Subtract the numerators.
Distribute the negative sign.
Combine like terms.
x for which x2 + 1 = 0;
defined.
5-3
Adding and Subtracting
Rational Expressions
Check It Out! Example 1a
Add or subtract. Identify any x-values for
which the expression is undefined.
6x + 5 + 3x – 1
x2 – 3
x2 – 3
6x + 5 + 3x – 1
x2 – 3
9x + 4
x2 – 3
Add the numerators.
Combine like terms.
The expression is undefined at x = ±
this value makes x2 – 3 equal 0.
Holt McDougal Algebra 2
because
5-3
Adding and Subtracting
Rational Expressions
Check It Out! Example 1b
Add or subtract. Identify any x-values for
which the expression is undefined.
3x2 – 5 – 2x2 – 3x – 2
3x – 1
3x – 1
3x2 – 5 – (2x2 – 3x – 2) Subtract the numerators.
3x – 1
3x2 – 5 – 2x2 + 3x + 2
Distribute the negative sign.
3x – 1
x2 + 3x – 3
Combine like terms.
3x – 1
1 because
The expression is undefined at x = 3
this value makes 3x – 1 equal 0.
Holt McDougal Algebra 2
5-3
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 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.
Holt McDougal Algebra 2
5-3
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 McDougal Algebra 2
Multiply.
5-3
Adding and Subtracting
Rational Expressions
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 ÷
x
x2 – 1
x–1
Write as division.
x+1 
x2 – 1
Multiply by the reciprocal.
x–1
x
Holt McDougal Algebra 2
5-3
Adding and Subtracting
Rational Expressions
Check It Out! Example 5a Continued
Simplify. Assume that all expressions are defined.
x+1
 x–1
(x – 1)(x + 1)
x
1
x
Holt McDougal Algebra 2
Factor the denominator.
Divide out common factors.
5-3
Adding and Subtracting
Rational Expressions
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 ÷
6
x–1
3x – 3
Write as division.
20  3x – 3
x–1
6
Multiply by the reciprocal.
Holt McDougal Algebra 2
5-3
Adding and Subtracting
Rational Expressions
Check It Out! Example 5b Continued
Simplify. Assume that all expressions are defined.
20  3(x – 1)
x–1
6
10
Holt McDougal Algebra 2
Factor the numerator.
Divide out common factors.
5-3
Adding and Subtracting
Rational Expressions
Essential Question
• How do you add rational expressions?
• To add or subtract rational expressions
with like denominators, add or subtract the
numerators and use the same
denominator.
Holt McDougal Algebra 2
5-3
Adding and Subtracting
Rational Expressions
Student: It was too complex.
Holt McDougal Algebra 2