angel, elementary and intermediate algebra, 3ed 1 rational expressions and equations chapter 7
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Angel, Elementary and Intermediate Algebra, 3ed 1
Rational Expressions
and Equations
Chapter 7
Angel, Elementary and Intermediate Algebra, 3ed 2
§ 7.1
Simplifying Rational
Expressions
Angel, Elementary and Intermediate Algebra, 3ed 3
Angel, Elementary and Intermediate Algebra, 3ed 4
Rational Expressions
A rational expression is an expression of the form p/q where p and q are polynomials and q 0.
Examples:
Whenever a rational expression contains a variable in the denominator, assume that the values that make the denominator 0 are excluded.
Angel, Elementary and Intermediate Algebra, 3ed 5
Signs of a FractionThree signs are associated with any fraction: the sign of the numerator, the sign of the denominator, and the sign of the fraction.
Changing any two of the three signs of a fraction does not change the value of the fraction
- a+ b
+
sign of the numerator
sign of the denominator
sign of the fraction
- a b
a -b
= a b
-=
Angel, Elementary and Intermediate Algebra, 3ed 6
Simplifying
A rational expression is simplified or reduced to lowest terms when the numerator and denominator have no common factors other than 1. Examples:
Angel, Elementary and Intermediate Algebra, 3ed 7
Angel, Elementary and Intermediate Algebra, 3ed 8
Simplifying Rational Expressions
1. Factor both the numerator and denominator as completely as possible.
2. Divide out any factors common to both the numerator and denominator.
Angel, Elementary and Intermediate Algebra, 3ed 9
Factoring a Negative 1
Remember that when –1 is factored from a polynomial, the sign of each term in the polynomial changes.
Example: – 2x + 5 = – 1(2x – 5) = –(2x – 5)
Angel, Elementary and Intermediate Algebra, 3ed 10
Angel, Elementary and Intermediate Algebra, 3ed 11
Angel, Elementary and Intermediate Algebra, 3ed 12
§ 7.2
Multiplication and Division of
Rational Expressions
Angel, Elementary and Intermediate Algebra, 3ed 13
Multiplying Fractions
0d and 0b,db
ca
d
c
b
a
Multiply . - 521
·34
- 521
·34
=- 521
·34
1
7
- 57
·14
= =- 528
Angel, Elementary and Intermediate Algebra, 3ed 14
Multiplying Rational Expressions
1. Factor all numerators and denominators completely.
2. Divide out common factors.3. Multiply numerators together and
multiply denominators together.
Multiply .yx
22z
11z
y18x-52
32
52
32
yx
22z
11z
y18x-4
2
y
36z
52
32
yx
22z
11z
y18x-
Angel, Elementary and Intermediate Algebra, 3ed 15
Angel, Elementary and Intermediate Algebra, 3ed 16
Dividing Two Fractions
0c and 0d , 0b ,bc
ad
c
d
b
a
d
c
b
a
Divide . - 2 9
59
=- 2 9
59
- 2 9
·95
1- 2 5
=
1
- 2 9
·95
=
Angel, Elementary and Intermediate Algebra, 3ed 17
Dividing Rational Expressions
Invert the divisor (the second fraction) and multiply
Divide .3017-x
1
127xx
122
1
3017-x
187xx
1
3017-x
1
187xx
1 2
222
1
15)2)(x(x
2)9)(x(x
1
9)(x
15)(x
Angel, Elementary and Intermediate Algebra, 3ed 18
Angel, Elementary and Intermediate Algebra, 3ed 19
Factor the x-box way
Example: Factor 3x2 -13x -10
-13x
(3x2)(-10)=
-30x2
-15x 2x
-10
-15x
2x
3x2
x -5
3x
+2
3x2 -13x -10 = (x-5)(3x+2)
Angel, Elementary and Intermediate Algebra, 3ed 20
Factor the x-box way
Example: Factor 2x2 +3x -9
3x
(2x2)(-9)=
-18x2
6x -3x
-9
6x
-3x
2x2
x +3
2x
-3
2x2 +3x -9=(x+3)(2x-3)
Angel, Elementary and Intermediate Algebra, 3ed 21
Factor the x-box way
Example: Factor 3x2 -2x -5
-2x
(3x2)(-5)=
-15x2
-5x 3x
-5
-5x
3x
3x2
3x -5
x
+1
3x2 -2x -5=(3x-5)(x+1)
Angel, Elementary and Intermediate Algebra, 3ed 22
Angel, Elementary and Intermediate Algebra, 3ed 23
§ 7.3
Addition and Subtraction of
Rational Expressions with a
Common Denominator
Angel, Elementary and Intermediate Algebra, 3ed 24
Adding/Subtracting Fractions
0c ,c
ba
c
b
c
a 0c ,
c
ba
c
b
c
a
712
= 512
212
+
Add . 512
212
+
Angel, Elementary and Intermediate Algebra, 3ed 25
Common Denominators
1. Add or subtract the numerators.2. Place the sum or difference of the
numerators found in step 1 over the common denominator.
3. Simplify the fraction if possible.
Subtract .5
6
5
7-2x
5
13-2x
5
6-7-2x
5
6
5
7-2x
Angel, Elementary and Intermediate Algebra, 3ed 26
Common Denominators
a.) Add .12ww
4-2w-
12ww
53w22
Example:
12ww
4-2w-53w
12ww
4-2w-
12ww
53w222
1)(w
1
1)(w
1w2
12ww
4-2w-53w2
Angel, Elementary and Intermediate Algebra, 3ed 27
Common Denominators
b.) Subtract
.649x
29x-x
649x
54x2
2
2
2
649x
29)x-(x-54x
649x
29x-x
649x
54x2
22
2
2
2
2
649x
24x3x
649x
29xx-54x2
2
2
22
8)(3x
3)(x
8)8)(3x(3x
8)3)(3x(x
Example:
Angel, Elementary and Intermediate Algebra, 3ed 28
Least Common Denominator
1. Factor each denominator completely. Any factors used more than once should be expressed as powers.
2. List all different factors that appear in any of the denominators. When the same factor appears in more than one denominator, write that factor with the highest power that appears.
3. The least common denominator (LCD) is the product of all the factors listed in step 2.
Angel, Elementary and Intermediate Algebra, 3ed 29
Least Common Denominator
a.)5-x
6
52x
3-x
Find the LCD:
The LCD is (2x + 5)(x – 5).
x
x
xx
2-5x 2
2
b.) The LCD is x(x + 1).x
x
1)x(x
2-5x 2
245 9wz
2
z4w
3c.) The LCD is 36w5z4
.
Angel, Elementary and Intermediate Algebra, 3ed 30
§ 7.4
Addition and Subtraction of
Rational Expressions
Angel, Elementary and Intermediate Algebra, 3ed 31
Unlike Denominators
1. Determine the LCD.2. Rewrite each fraction as an
equivalent fraction with the LCD.3. Add or subtract the numerators
while maintaining the LCD.4. When possible, factor the
remaining numerator and simplify the fraction.
Angel, Elementary and Intermediate Algebra, 3ed 32
Unlike Denominators
a.)w
5
2w
3
2w
2w
w
5
w
w
2w
3
The LCD is w(w+2).
2)w(w
2)5(w
2)w(w
3w
2)w(w
105w
2)w(w
3w
2)w(w
108w
answers. acceptable also are and 2ww
108w
2)w(w
5)2(4w2
Example:
Angel, Elementary and Intermediate Algebra, 3ed 33
Unlike Denominators
b.)3x
1
4-4x
x The LCD is 12x(x – 1).
3x
1
1)-4(x
x
1)-4(x
1)-4(x
3x
1
3x
3x
1)-4(x
x
1)-12x(x
44x3x
1)-12x(x
1)-4(x
1)-12x(x
3x 22
This cannot be factored any further.
Example:
Angel, Elementary and Intermediate Algebra, 3ed 34
Unlike Denominators
c.)5-x
2
10-3x-x
3x2
The LCD is
(x – 5)(x+2).5-x
2
2)5)(x-(x
3x
2x
2x
5-x
2
2)5)(x-(x
3x
2)5)(x-(x
2)2(x
2)5)(x-(x
3x
2)5)(x-(x
42x
2)5)(x-(x
3x
2)5)(x-(x
4)(2x-3x
2)5)(x-(x
42x3x
2)5)(x-(x
1x
Example:
Angel, Elementary and Intermediate Algebra, 3ed 35
§ 7.5
Complex Fractions
Angel, Elementary and Intermediate Algebra, 3ed 36
Simplifying Complex Fractions
A complex fraction is one that has a fraction in its numerator or its denominator or in both the numerator and denominator.
454
3xx
3x
ba9-a
ba
Example:
Angel, Elementary and Intermediate Algebra, 3ed 37
Simplify by Combining Like Terms
1. Add or subtract the fraction in both the numerator and denominator of the complex fraction to obtain single fractions in both the numerator and the denominator.
2. Invert the denominator of the complex fraction and multiply the numerator by it.
3. Simplify further if possible.
Angel, Elementary and Intermediate Algebra, 3ed 38
4
b
b
12a 2
3
4bb
12a
2
3a)
23 b
4
b
12a
5b
48a
yx
y-xx
2b
) y
x
y-x
x 2
2x
y
y-x
x
y)-(xx
xy2
Simplify by Combining Like Terms
y)-x(x
y
Simplify:
Angel, Elementary and Intermediate Algebra, 3ed 39
Simplify by Multiplying
1. Find the LCD of all the denominators
appearing in the complex fraction.
2. Multiply both the numerator and the
denominator of the complex fraction by
the LCD.
3. Simplify further if possible.
Angel, Elementary and Intermediate Algebra, 3ed 40
Simplify by Multiplying
4bb
12a
2
3a)
LCD is 4b3.
4b3
4b
b12a
2
3
4b3
44bb
48ab
5
3
3
5b
48a
yx
y-xx
2b
)LCD is y(x-y).
y(x-y)
y(x-y)yx
y-xx
2
yy)-y(xx
y-xy)-xy(x
2y)-(xx
xy2
y)-x(x
y
Simplify:
Angel, Elementary and Intermediate Algebra, 3ed 41
§ 7.6
Solving Rational Equations
Angel, Elementary and Intermediate Algebra, 3ed 42
Complex Fractions
A rational equation is one that contains one or more rational (fractional) expressions.
8x5
3x
2
1 5
3x
3
Example:
Angel, Elementary and Intermediate Algebra, 3ed 43
Solving Rational Equations
1. Find the LCD of all fractions in the equation.
2. Multiply both sides of the equation by the LCD. (Every term will be multiplied by the LCD.)
3. Remove any parentheses and combine like terms on each side of the equation.
4. Solve the equation.
5. Check the solution in the original equation.
Angel, Elementary and Intermediate Algebra, 3ed 44
Integer Denominators
46
x
5
xa
)The LCD is 30. 304
6
x
5
x30
1206
30x
5
30x 1205x6x 120x
CHECK: 46
120
5
120 42024
Solve the equation:
Angel, Elementary and Intermediate Algebra, 3ed 45
Variable Denominators
The LCD is 2z.Solve: 2
11
z
35
CHECK:
Whenever there is a variable in the denominator, it is necessary to check your answer in the original equation. If the answer obtained makes the denominator zero, that value is NOT a solution to the equation.
2
112z
z
352z
2
22z
z
6z10z 11z610z z6
2
11
z
35
2
11
2
15
2
11
6
35
Angel, Elementary and Intermediate Algebra, 3ed 46
Variable Denominators
The LCD is 2(x-3).Solve:
3-x
3
2
3
3-x
x
CHECK:
3-x
33)2(x
2
3
3-x
x3)2(x
3-x
3)-6(x
2
3)-6(x
3-x
3)-2x(x 63)-3(x2x
693x2x 155x 3x
3-3
3
2
3
3-3
3
0
3
2
3
0
3
No solution
Angel, Elementary and Intermediate Algebra, 3ed 47
§ 7.7
Rational Equations:
Applications & Problem Solving
Angel, Elementary and Intermediate Algebra, 3ed 48
Work Problems
Problems in which two or more people or machines work together to complete a certain task are referred to as work problems.
part of task done by first machine
part of task done by second machine
1 (one whole task
completed)+ =
Angel, Elementary and Intermediate Algebra, 3ed 49
Work Problems
At the NCNB Savings Bank it takes a computer 4 hours to process and print payroll checks. When a second computer is used and the two computers work together, the checks can be printed in 3 hours. How long would it take the second computer by itself to process and print the payroll checks?
Example:
Continued.
Angel, Elementary and Intermediate Algebra, 3ed 50
Work Problems
Machine
Rate of Work Time
Part of Task
(rate x time)
1st
2nd
A table helps to keep information
organized.
Example continued:
Continued.
Angel, Elementary and Intermediate Algebra, 3ed 51
Work Problems
Machine
Rate of Work Time
Part of Task
(rate x time)
1st
2nd
4
1
x
1
4
3
x
3
1x
3
4
3 One job done
3
3
Solve the equation.
Example continued:
Continued.
Angel, Elementary and Intermediate Algebra, 3ed 52
Work Problems
1x
3
4
3 The LCD is 4x.
1
x
3
4
34x 4x123x x12
It would take the second computer 12 hours by itself.
CHECK: 112
3
4
3 1
12
3
12
9
Example continued: