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Holt Algebra 2
8-3 Adding and Subtracting Rational Expressions8-3 Adding and Subtracting Rational Expressions
Holt Algebra 2
Warm Up
Lesson Presentation
Lesson Quiz
Holt Algebra 2
8-3 Adding and Subtracting Rational Expressions
Warm UpAdd or subtract.
1.
2.
3.
4.
x ≠ 0
1324
x ≠ –1, x ≠ 1
Simplify. Identify any x-values for which the expression is undefined.
1315
1112
3 8–
715
2 5 +
x – 1x2 – 1
1x + 1
1 3 x6
4x9
12x3
Holt Algebra 2
8-3 Adding and Subtracting Rational Expressions
Add and subtract rational expressions.
Simplify complex fractions.
Objectives
Holt Algebra 2
8-3 Adding and Subtracting Rational Expressions
complex fraction
Vocabulary
Holt Algebra 2
8-3 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 Algebra 2
8-3 Adding and Subtracting Rational Expressions
Add or subtract. Identify any x-values for which the expression is undefined.
Example 1A: Adding and Subtracting Rational Expressions with Like Denominators
Add the numerators.
The expression is undefined at x = –4 because this value makes x + 4 equal 0.
x – 3 x + 4
+ x – 2x + 4
x – 3 +x + 4
x – 2
2x – 5x + 4
Combine like terms.
Holt Algebra 2
8-3 Adding and Subtracting Rational Expressions
Add or subtract. Identify any x-values for which the expression is undefined.
Example 1B: Adding and Subtracting Rational Expressions with Like Denominators
Subtract the numerators.
There is no real value of x for which x2 + 1 = 0; the expression is always defined.
3x – 4 x2 + 1
– 6x + 1x2 + 1
3x – 4 –x2 + 1 (6x + 1)
–3x – 5x2 + 1 Combine like terms.
Distribute the negative sign. 3x – 4 –
x2 + 1 6x – 1
Holt Algebra 2
8-3 Adding and Subtracting Rational Expressions
Check It Out! Example 1a
Add or subtract. Identify any x-values for which the expression is undefined.
Add the numerators.
6x + 5 x2 – 3
+ 3x – 1x2 – 3
6x + 5 +x2 – 3
3x – 1
9x + 4x2 – 3
Combine like terms.
The expression is undefined at x = ± because this value makes x2 – 3 equal 0.
Holt Algebra 2
8-3 Adding and Subtracting Rational Expressions
Check It Out! Example 1b
Add or subtract. Identify any x-values for which the expression is undefined.
Subtract the numerators.
3x2 – 5 3x – 1
– 2x2 – 3x – 23x – 1
3x – 1 3x2 – 5 – (2x2 – 3x – 2)
x2 + 3x – 33x – 1 Combine like terms.
Distribute the negative sign. 3x2 – 5 –
3x – 1 2x2 + 3x + 2
The expression is undefined at x = because this value makes 3x – 1 equal 0.
1 3
Holt Algebra 2
8-3 Adding and Subtracting Rational Expressions
To add or subtract rational expressions with unlike denominators, first find the least common denominator (LCD). The LCD is the least common multiple of the polynomials in the denominators.
Holt Algebra 2
8-3 Adding and Subtracting Rational Expressions
Find the least common multiple for each pair.
Example 2: Finding the Least Common Multiple of Polynomials
A. 4x2y3 and 6x4y5
4x2y3 = 2 2 x2 y3
6x4y5 = 3 2 x4 y5
The LCM is 2 2 3 x4 y5, or 12x4y5.
B. x2 – 2x – 3 and x2 – x – 6
x2 – 2x – 3 = (x – 3)(x + 1)
x2 – x – 6 = (x – 3)(x + 2)
The LCM is (x – 3)(x + 1)(x + 2).
Holt Algebra 2
8-3 Adding and Subtracting Rational Expressions
Check It Out! Example 2
Find the least common multiple for each pair.
a. 4x3y7 and 3x5y4
4x3y7 = 2 2 x3 y7
3x5y4 = 3 x5 y4
The LCM is 2 2 3 x5 y7, or 12x5y7.
b. x2 – 4 and x2 + 5x + 6
x2 – 4 = (x – 2)(x + 2)
x2 + 5x + 6 = (x + 2)(x + 3)
The LCM is (x – 2)(x + 2)(x + 3).
Holt Algebra 2
8-3 Adding and Subtracting Rational Expressions
To add rational expressions with unlike denominators, rewrite both expressions with the LCD. This process is similar to adding fractions.
Holt Algebra 2
8-3 Adding and Subtracting Rational Expressions
Add. Identify any x-values for which the expression is undefined.
Example 3A: Adding Rational Expressions
x – 3 x2 + 3x – 4
+ 2x x + 4
x – 3 (x + 4)(x – 1)
+ 2x x + 4 Factor the denominators.
The LCD is (x + 4)(x – 1), so multiply by . 2x
x + 4
x – 1 x – 1
x – 3 (x + 4)(x – 1)
+ 2x x + 4
x – 1 x – 1
Holt Algebra 2
8-3 Adding and Subtracting Rational Expressions
x – 3 + 2x(x – 1) (x + 4)(x – 1)
Add the numerators.
2x2 – x – 3 (x + 4)(x – 1)
Simplify the numerator.
Example 3A Continued
Add. Identify any x-values for which the expression is undefined.
Write the sum in factored or expanded form.
2x2 – x – 3 (x + 4)(x – 1)
2x2 – x – 3 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.
Holt Algebra 2
8-3 Adding and Subtracting Rational Expressions
Add. Identify any x-values for which the expression is undefined.
x x + 2
+ –8 x2 – 4
x x + 2
+ –8(x + 2)(x – 2) Factor the denominator.
The LCD is (x + 2)(x – 2), so multiply by . x
x + 2
x – 2 x – 2
Example 3B: Adding Rational Expressions
x – 2 x – 2
x x + 2
+ –8(x + 2)(x – 2)
x(x – 2) + (–8) (x + 2)(x – 2)
Add the numerators.
Holt Algebra 2
8-3 Adding and Subtracting Rational Expressions
x2 – 2x – 8 (x + 2)(x – 2)
Write the numerator in standard form.
Add. Identify any x-values for which the expression is undefined.
Example 3B Continued
(x + 2)(x – 4)(x + 2)(x – 2)
Factor the numerator.
x – 4x – 2
Divide out common factors.
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.
Holt Algebra 2
8-3 Adding and Subtracting Rational Expressions
Check It Out! Example 3a
Add. Identify any x-values for which the expression is undefined.
3x 2x – 2
+ 3x – 2 3x – 3
Factor the denominators.
The LCD is 6(x – 1), so
multiply by 3 and
by 2.
3x 2(x – 1)
3x – 2 3(x – 1)
3x 2(x – 1)
+ 3x – 2 3(x – 1)
3 3
3x 2(x – 1)
+ 3x – 2 3(x – 1)
2 2
Holt Algebra 2
8-3 Adding and Subtracting Rational Expressions
Check It Out! Example 3a Continued
Add. Identify any x-values for which the expression is undefined.
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.
Add the numerators.9x + 6x – 4 6(x – 1)
Holt Algebra 2
8-3 Adding and Subtracting Rational Expressions
Add. Identify any x-values for which the expression is undefined.
2x + 6 x2 + 6x + 9
+x x + 3
Factor the denominators.
Check It Out! Example 3b
2x + 6 (x + 3)(x + 3)
+x x + 3
The LCD is (x + 3)(x + 3), so
multiply by . x
(x + 3) (x + 3) (x + 3)
2x + 6 (x + 3)(x + 3)
+x x + 3
x + 3 x + 3
Add the numerators.x2 + 3x + 2x + 6 (x + 3)(x + 3)
Holt Algebra 2
8-3 Adding and Subtracting Rational Expressions
Check It Out! Example 3b Continued
Add. Identify any x-values for which the expression is undefined.
Write the numerator in standard form.
(x + 3)(x + 2)(x + 3)(x + 3)
Factor the numerator.
x + 2x + 3
Divide out common factors.
The expression is undefined at x = –3 because this value of x make the factors (x + 3) and (x + 3) equal 0.
x2 + 5x + 6 (x + 3)(x + 3)
Holt Algebra 2
8-3 Adding and Subtracting Rational Expressions
Example 4: Subtracting Rational Expressions
Factor the denominators.
Subtract . Identify any x-
values for which the expression is undefined.
2x2 – 30x2 – 9
–x + 3x + 5
2x2 – 30(x – 3)(x + 3)
–x + 3x + 5
The LCD is (x – 3)(x + 3),
so multiply by . x + 5 x + 3
(x – 3) (x – 3)
2x2 – 30(x – 3)(x + 3)
–x + 3x + 5 x – 3
x – 3
2x2 – 30 – (x + 5)(x – 3)(x – 3)(x + 3)
Subtract the numerators.
2x2 – 30 – (x2 + 2x – 15)(x – 3)(x + 3)
Multiply the binomials in the numerator.
Holt Algebra 2
8-3 Adding and Subtracting Rational Expressions
Example 4 Continued
Subtract . Identify any x-
values for which the expression is undefined.
2x2 – 30x2 – 9
–x + 3x + 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 – 5x – 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.
Holt Algebra 2
8-3 Adding and Subtracting Rational Expressions
Subtract . Identify any x-
values for which the expression is undefined.
3x – 22x + 5
–5x – 2 2
(3x – 2)(5x – 2) – 2(2x + 5)(2x + 5)(5x – 2)
Subtract the numerators.
15x2 – 16x + 4 – (4x + 10)(2x + 5)(5x – 2)
Multiply the binomials in the numerator.
Check It Out! Example 4a
3x – 22x + 5
–5x – 2 2 2x + 5
2x + 5 5x – 2 5x – 2
The LCD is (2x + 5)(5x – 2),
so multiply by
and by .
3x – 2 2x + 5
(5x – 2) (5x – 2)
2 5x – 2
(2x + 5) (2x + 5)
Holt Algebra 2
8-3 Adding and Subtracting Rational Expressions
Check It Out! Example 4a Continued
Distribute the negative sign.
The expression is undefined at x = – and x = because these values of x make the factors (2x + 5) and (5x – 2) equal 0.
15x2 – 16x + 4 – 4x – 10(2x + 5)(5x – 2)
5 2
2 5
Subtract . Identify any x-
values for which the expression is undefined.
3x – 22x + 5
–5x – 2 2
Holt Algebra 2
8-3 Adding and Subtracting Rational Expressions
Check It Out! Example 4b
Subtract . Identify any x-
values for which the expression is undefined.
2x2 + 64x2 – 64
–x + 8 x – 4
2x2 + 64 – (x – 4)(x – 8)(x – 8)(x + 8)
Subtract the numerators.
2x2 + 64 – (x2 – 12x + 32)(x – 8)(x + 8)
Multiply the binomials in the numerator.
Factor the denominators. 2x2 + 64
(x – 8)(x + 8)–
x + 8x – 4
The LCD is (x – 3)(x + 8),
so multiply by . x – 4 x + 8
(x – 8) (x – 8)
2x2 + 64(x – 8)(x + 8)
–x + 8x – 4 x – 8
x – 8
Holt Algebra 2
8-3 Adding and Subtracting Rational Expressions
Check It Out! Example 4b
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 + 4x – 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.
Subtract . Identify any x-
values for which the expression is undefined.
2x2 + 64x2 – 64
–x + 8 x – 4
2x2 + 64 – x2 + 12x – 32)(x – 8)(x + 8)
Holt Algebra 2
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 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 Algebra 2
8-3 Adding and Subtracting Rational Expressions
Example 5A: Simplifying Complex Fractions
Simplify. Assume that all expressions are defined.
Write as division.
x + 2x – 1x – 3x + 5
Write the complex fraction as division.
x + 2x – 1
x – 3x + 5
÷
Multiply by the reciprocal.
x + 2x – 1
x + 5x – 3
Multiply.x2 + 7x + 10x2 – 4x + 3
(x + 2)(x + 5)(x – 1)(x – 3)
or
Holt Algebra 2
8-3 Adding and Subtracting Rational Expressions
Example 5B: Simplifying Complex Fractions
Simplify. Assume that all expressions are defined.
The LCD is 2x.
Multiply the numerator and denominator of the complex fraction by the LCD of the fractions in the numerator and denominator.
x – 1x
x2
3x +
x – 1x
(2x)
x2
3x + (2x)(2x)
Holt Algebra 2
8-3 Adding and Subtracting Rational Expressions
Divide out common factors.
Simplify.x2 + 6
2(x – 1)
(3)(2) + (x)(x)(x – 1)(2)
Example 5B Continued
Simplify. Assume that all expressions are defined.
x2 + 62x – 2
or
Holt Algebra 2
8-3 Adding and Subtracting Rational Expressions
Simplify. Assume that all expressions are defined.
Write as division.
x + 1x2 – 1
x x – 1
Write the complex fraction as division.
x + 1x2 – 1
xx – 1
÷
Multiply by the reciprocal.x + 1x2 – 1
x – 1x
Check It Out! Example 5a
Holt Algebra 2
8-3 Adding and Subtracting Rational Expressions
1x
Simplify. Assume that all expressions are defined.
Check It Out! Example 5a Continued
x + 1(x – 1)(x + 1)
x – 1x
Divide out common factors.
Factor the denominator.
Holt Algebra 2
8-3 Adding and Subtracting Rational Expressions
Check It Out! Example 5b
Simplify. Assume that all expressions are defined.
Write as division.
20x – 1
63x – 3
Write the complex fraction as division.
20x – 1
63x – 3
÷
Multiply by the reciprocal.20x – 1
3x – 36
Holt Algebra 2
8-3 Adding and Subtracting Rational Expressions
Check It Out! Example 5b Continued
Simplify. Assume that all expressions are defined.
Divide out common factors.
20x – 1
3(x – 1)6
Factor the numerator.
10
Holt Algebra 2
8-3 Adding and Subtracting Rational Expressions
Check It Out! Example 5c
Simplify. Assume that all expressions are defined.
The LCD is (2x)(x – 2).
Multiply the numerator and denominator of the complex fraction by the LCD of the fractions in the numerator and denominator.
x + 4x – 2
12x
1x +
x + 4x – 2
(2x)(x – 2)
12x
1x + (2x)(x – 2)(2x)(x – 2)
Holt Algebra 2
8-3 Adding and Subtracting Rational Expressions
Check It Out! Example 5c Continued
Divide out common factors.
Simplify.3x – 6
(x + 4)(2x)
(2)(x – 2) + (x – 2)(x + 4)(2x)
3(x – 2)2x(x + 4)
or
Simplify. Assume that all expressions are defined.
Holt Algebra 2
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.
d0.8
d1.4+Total time: Use the formula t = .
rd
d0.8
d1.4+
2dAverage speed: The average speed is . total timetotal distance
Holt Algebra 2
8-3 Adding and Subtracting Rational Expressions
d0.8
d1.4+
2d = = and = = . 1.4
d d
57 7
5d0.8d d
54 4
5d
5d 4
5d 7 +
2d(28)
(28) (28)The LCD of the fractions in the denominator is 28.
56d20d + 35d
Combine like terms and divide out common factors.
Simplify.
55d55d
≈ 1.0
The hiker’s average speed is 1.0 mi/h.
Example 6 Continued
Holt Algebra 2
8-3 Adding and Subtracting Rational Expressions
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.
d45
d40 +Total time: Use the formula t = .
rd
d45
d40 +
2dAverage speed: The average speed is . total timetotal distance
Check It Out! Example 6
Holt Algebra 2
8-3 Adding and Subtracting Rational Expressions
d40 (360) d
45+
2d(360)
(360)The LCD of the fractions in the denominator is 360.
720d9d + 8d
Combine like terms and divide out common factors.
Simplify.
720d17d
≈ 42.4
Justin’s average speed is 42.4 mi/h.
Check It Out! Example 6
Holt Algebra 2
8-3 Adding and Subtracting Rational Expressions
Lesson Quiz: Part I
2.
3x2 – 2x + 7 (x – 2)(x + 1)
x ≠ –1, 2
x – 9 x – 4
x ≠ 4, –4
Add or subtract. Identify any x-values for which the expression is undefined.
1. 2x + 1 x – 2
+ x – 3x + 1
x x + 4
– x + 36x2 – 16
3. Find the least common multiple of x2 – 6x + 5 and x2 + x – 2.
(x – 5)(x – 1)(x + 2)
Holt Algebra 2
8-3 Adding and Subtracting Rational Expressions
Lesson Quiz: Part II
1 x
48 mi/h
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?
4. Simplify . Assume that all expressions are defined.
x + 2x2 – 4
x x – 2
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