8-3: Adding and Subtracting Rational Expressions
goals:
•Add or subtract rational expressions and identify values for which the expression is
undefined.
•Simplify compound rational expressions
Standard: 9.2.3.4
When adding or subtracting fractions, you must find the least
common denominator (LCD). The LCD is the least common
multiple of both denominators.
APK
Find the sum of the following fractions.
2 7
+
5 15
Factor the denominators to find the LCD.
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2
7
+
5 3⋅ 5
Multiply by “1” to get common denominators.
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€
2 3
7
⋅ +
5 3 3⋅ 5
6 7
+
15 15
Add the numerators.
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€
13
15
Find the difference of the following fractions.
11 3
−
12 8
€
Change the subtraction to “adding the opposite.”
Factor the denominators to find the LCD.
11
−3
+
2⋅ 2⋅ 3 2⋅ 2⋅ 2
Multiply by “1” to get common denominators.
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€
11 2
−3 3
⋅ +
⋅
2⋅ 2⋅ 3 2 2⋅ 2⋅ 2 3
22 −9
+
24 24
Add the numerators.
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NEW INFORMATION
How do you add or
subtract rational
expressions with like
denominators?
13
24
Adding rational expressions is similar to adding fractions. To do
so, you must have a common denominator. To simplify, add the
like terms in the numerator and keep the denominator. When
subtracting, change the sign of each term in the second numerator
and change to addition (“adding the opposite”).
Add or subtract. Identify any x-values for which the expression
is undefined.
Example 1
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2x − 3 3x + 4
+
4 x +1 4 x +1
The denominators are the same.
2x − 3 + 3x + 4
4 x +1
Add the numerators.
5x +1
4 x +1
Simplify.
1
Since 4 x +1 ≠ 0 , x ≠ − .
4
€ Example 2
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€
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3x − 4 5x + 3
−
4x + 5 4x + 5
The denominators are the same.
3x − 4 −5x − 3
+
4x + 5 4x + 5
Distribute –1.
3x − 4 − 5x − 3
4x +5
Add the numerators.
−2x − 7
4x + 5
Simplify.
5
Since 4 x + 5 ≠ 0 , x ≠ − .
4
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€ Example 3
€
4x − 3 4x + 3
−
2x − 5 2x − 5
€
€
€
The denominators are the same.
4 x − 3 −4 x − 3
+
2x − 5 2x − 5
Distribute –1.
4x − 3 − 4x − 3
2x − 5
Add the numerators.
−6
2x − 5
Simplify.
5
Since 2x − 5 ≠ 0 , x ≠ .
2
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How do you find the LCM
of polynomials?
Finding€the least common multiple of polynomials.
1. Completely factor the polynomials.
2. Determine common factors.
3. Multiply each polynomial by the factors in the other polynomial.
Find the least common multiple for each pair.
Example 1
4 x 2 y 3 and 16x 4 y
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4 x 2 y 3 = 2⋅ 2⋅ x⋅ x⋅ y⋅ y⋅ y
16x 4 y = 2⋅ 2⋅ 2⋅ 2⋅ x⋅ x⋅ x⋅ x⋅ y
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LCM is 2⋅ 2⋅ 2⋅ 2⋅ x⋅ x⋅ x⋅ x⋅ y⋅ y⋅ y = 16x 4 y 3 .
Example 2
€
x 2 − 25 and x 2 +10x + 25
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€
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How do you add or
subtract rational
expressions with unlike
denominators?
x 2 − 25 = (x + 5)(x − 5)
x 2€+10x + 25 = (x + 5)(x + 5)
LCM is (x + 5)(x + 5)(x − 5) = (x + 5) 2 (x − 5) .
To add rational expressions with unlike terms, you must find a
€common denominator.
1.
2.
3.
4.
Completely factor the denominators.
Determine the Least Common Denominator (LCD).
Multiply by “1” to change the fractions.
Add the numerators.
Add or subtract. Identify any x-value for which the expression
is undefined.
Example 1
3x − 2 2x − 3
+
x + 6 2x −1
€
Determine the LCD.
Multiply by “1” to change the fractions.
(3x − 2) (2x −1) (2x − 3) (x + 6)
⋅
+
⋅
(x + 6) (2x −1) (2x −1) (x + 6)
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6x 2 − 7x + 2
2x 2 + 9x −18
+
(x + 6)(2x −1) (2x −1)(x + 6)
Add the numerators and simplify.
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8x 2 + 2x −16
(x + 6)(2x −1)
€
2(4 x 2 + x − 8)
(x + 6)(2x −1)
x ≠ −6 , x ≠
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€
1
2
Example 2
€
4 x − 5 3x −1
+
12x + 4 3x +1
Completely factor the denominators and determine the LCD.
€
4x − 5
3x −1
+
4(3x +1) 3x +1
Multiply by “1” to change the fractions.
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4x − 5
(3x −1) 4
+
⋅
4(3x +1) (3x +1) 4
4x − 5
12x − 4
+
4(3x +1) 4(3x +1)
Add the numerators and simplify.
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16x − 9
4(3x +1)
x≠−
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€
1
3
Example 3
3x − 4 2x −1
+
x2 − 9 x + 3
Completely factor the denominators and determine the LCD.
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3x − 4
2x −1
+
(x + 3)(x − 3) (x + 3)
Multiply by “1” to change the fractions.
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3x − 4
(2x −1) (x − 3)
+
⋅
(x + 3)(x − 3) (x + 3) (x − 3)
3x − 4
2x 2 − 7x + 3
+
(x + 3)(x − 3) (x + 3)(x − 3)
Add the numerators and simplify.
2x 2 − 4 x −1
(x + 3)(x − 3)
x ≠ −3 , x ≠ 3
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Example 4
€
3x − 5 2x − 5
−
2x − 5 3x +1
Change subtraction to “adding the opposite.”
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€
3x − 5 −(2x − 5)
+
2x − 5
3x +1
Completely factor the denominators and determine the LCD.
Multiply by “1” to change the fractions.
(3x − 5) (3x +1) −(2x − 5) (2x − 5)
⋅
+
⋅
(2x − 5) (3x +1) (3x +1) (2x − 5)
€
9x 2 −12x − 5 −4 x 2 +10x − 25
+
(2x − 5)(3x +1) (2x − 5)(3x +1)
Add the numerators and simplify.
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5x 2 − 2x − 30
(2x − 5)(3x +1)
5
1
x≠ , x≠
2
3
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Example 5
2x + 8
3
−
2
x −16 x − 4
€
Change subtraction to “adding the opposite.”
Completely factor the denominators and determine the LCD.
2x + 8
−3
+
(x − 4)(x + 4) x − 4
Multiply by “1” to change the fractions.
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2x + 8
−3 (x + 4)
+
⋅
(x − 4)(x + 4) (x − 4) (x + 4)
2x + 8
−3x −12
+
(x − 4)(x + 4) (x − 4)(x + 4)
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Add the numerators and simplify.
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−x − 4
(x − 4)(x + 4)
−1(x + 4)
(x − 4)(x + 4)
€
−1
(x − 4)
€
−1
x −4
€
x ≠ −4 , x ≠ 4
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€
€
Example 6
x +2
x +1
−
x + 4x + 3 x + 3
2
€
Change subtraction to “adding the opposite.”
Completely factor the denominators and determine the LCD.
x +2
−(x +1)
+
(x + 3)(x +1)
x+3
Multiply by “1” to change the fractions.
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x +2
−(x +1) (x +1)
+
⋅
(x + 3)(x +1) (x + 3) (x +1)
x +2
−x 2 − 2x −1
+
(x + 3)(x +1) (x + 3)(x +1)
€
Add the numerators and simplify.
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−x 2 − x +1
(x + 3)(x +1)
−1(x 2 + x −1)
(x + 3)(x +1)
€
x ≠ −3 , x ≠ −1
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How do you simplify
complex fractions?
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Simplifying Complex Fractions
1.
2.
3.
4.
5.
Write the complex fraction as a division problem.
Multiply by the reciprocal.
Factor each numerator and each denominator.
Cancel common factors.
Simplify
Simplify. Assume that all expressions are defined.
Example 1
2x − 3
x −2
4x − 3
x2 − 4
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2x − 3 4 x − 3
Write the complex fraction as a division problem.
÷
x − 2 x2 − 4
2x − 3 x 2 − 4
⋅
x −2 4x − 3
Multiply by the reciprocal.
(2x − 3) (x − 2)(x + 2)
⋅
(x − 2)
(4 x − 3)
(2x − 3)(x + 2)
(4 x − 3)
Factor each num. and each den.
Cancel common factors.
Example 2
3x − 7
4x + 5
6x −1
5x − 6
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€
€
€
3x − 7 6x −1
Write the complex fraction as a division problem.
÷
4 x + 5 5x − 6
3x − 7 5x − 6
⋅
4 x + 5 6x −1
(3x − 7)(5x − 6)
(4 x + 5)(6x −1)
Example 3
Multiply by the reciprocal.
Remultiply.
2 1
+
x x
2x
x +2
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How can you calculate
average speed?
3
x
2x
x +2
Simplify the numerator.
3
2x
÷
x x +2
Write the complex fraction as a division problem.
3 x +2
⋅
x 2x
Multiply by the reciprocal.
3(x + 2)
2x 2
Remultiply.
Application Problem
Track Yvette ran at an average speed of 6.20 ft/s during the first
two laps of a race and an average speed of 7.75 ft/s during the
second two laps of a race. What was Yvette’s average speed for the
entire race? Round to the nearest tenth.
Let d represent the one-way distance.
Total distance is 2d.
Time is distance divided by speed.
Yvette’s time during the first two laps is
second two laps is
d
.
7.75
d
d€
Yvette’s total time is
.
+
6.20 7.75
€
2d
The average speed is
d
d
+
€
6.20 7.75
€
d
. Her time during the
6.20
2d(48.05)
d
d
(48.05) +
(48.05)
6.20
7.75
€
€
The LCD of the fractions is 48.05.
96.1d
7.75d + 6.20d
96.1d
≈ 6.9 ft/s.
13.95d
€
Summary
•Know how to add and subtract rational expressions.
•Know how to simplify complex rational expressions.
Other skills needed: factoring polynomials, adding/subtracting fractions.