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Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide R-1
Basic Concepts
of Algebra
Chapter R
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
R.1
The Real-Number System




Identify various kinds of real numbers.
Use interval notation to write a set of numbers.
Identify the properties of real numbers.
Find the absolute value of a real number.
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Rational Numbers


Numbers that can be expressed in the form p/q, where
p and q are integers and q  0.
Decimal notation for rational numbers either
terminates (ends) or repeats.
Examples:
1
a) 0
b)
8
c) 9
d)
7
11
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide R-4
Irrational Numbers


The real numbers that are not rational are irrational
numbers.
Decimal notation for irrational numbers neither
terminates nor repeats.
Examples:
a) 7.123444443443344…
b)
5
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Slide R-5
Interval Notation
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Slide R-6
Examples
Write interval notation for each set and graph the set.
a) {x|5 < x < 2}
Solution: {x|5 < x < 2} = (5, 2)
(
)
b) {x|4 < x  3}
Solution: {x|4 < x  3} = (4, 3]
(
]
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide R-7
Properties of the Real Numbers




Commutative property
a + b = b + a and ab = ba
Associative property
a + (b + c) = (a + b) + c and
a(bc) = (ab)c
Additive identity property
a+0=0+a=a
Additive inverse property
a + a = a + (a) = 0
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Slide R-8
More Properties


Multiplicative identity property
a•1=1•a=a
Multiplicative inverse property
1 1
a  a 1
a a

(a  0)
Distributive property
a(b + c) = ab + ac
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide R-9
Examples
State the property being illustrated in each sentence.
a) 7(6) = 6(7)
Commutative
b) 3d + 3c = 3(d + c)
Distributive
c) (3 + y) + x = 3 + (y + x)
Associative
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide R-10
Absolute Value

The absolute value of a number a, denoted |a|, is its
distance from 0 on the number line.
Example:
Simplify.
|6| = 6
|19| = 19
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Slide R-11
Distance Between Two Points on the
Number Line

For any real numbers a and b, the distance between a
and b is |a  b|, or equivalently |b  a|.
Example:
Find the distance between 4 and 3.
Solution:
The distance is |4  3| = |7| = 7,
or |3 (4)| = |3 + 4| = |7| = 7.
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide R-12
R.2
Integer Exponents, Scientific
Notation, and Order of Operations



Simplify expressions with integer exponents.
Solve problems using scientific notation.
Use the rules for order of operations.
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Integers as Exponents

When a positive integer is used as an exponent, it indicates the number
of times a factor appears in a product.
For any positive integer n,
a
n
 a  a  a  a  ...  a
base
n factors
where a is the base and n is the exponent.
Example: 84 = 8 • 8 • 8 • 8

For any nonzero real number a and any integer m,
m
a0 = 1 and a  1m .
a
Example: a) 80 = 1
b)
x 2
1
1
y5
2
5
 x  5  2  y  2
5
y
y
x
x
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide R-14
Properties of Exponents

Product rule
a a  a
m

n
Power rule
(am)n = amn
Raising a product to a power
mn
(ab)m = ambm
Quotient rule
am
mn

a
an



(a  0)
Raising a quotient to a power
m
am
a
   m
b
b
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(b  0)
Slide R-15
Examples – Simplify.
a) r 2 • r 5
r (2 + 5) = r 3
36 y 9 36 9  4
5
b)

y

2
y
18 y 4 18
c) (p6)4 = p -24 or 1
p24
d) (3a3)4 = 34(a3)4
= 81a12 or 81
a12
3
3
2 2
2 2



21
a
b
3
a
b 
e)

   4 
4
 7c

 c

33 a 6b 6 a 6b 6


12
c
27c12
a6

27b 6c12
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide R-16
Scientific Notation

Use scientific notation to name very large and very
small positive numbers and to perform computations.
Scientific notation for a number is an expression of
the type N  10m,
where 1  N < 10, N is in decimal notation, and m is an
integer.
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide R-17
Examples
Convert to scientific notation.
a) 17,432,000 = 1.7432  107
b) 0.00000000024 = 2.4  1010
Convert to decimal notation.
a) 3.481  106 = 3,481,000
b) 5.874  105 = 0.00005874
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide R-18
Another Example
Chesapeake Bay Bridge-Tunnel. The 17.6-mile-long
tunnel was completed in 1964. Construction costs were
$210 million. Find the average cost per mile.
2.1  108
8 1
 1.19  10
1
1.76  10
7
 $1.19  10
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide R-19
Rules for Order of Operations




Do all calculations within grouping symbols before
operations outside. When nested grouping symbols are
present, work from the inside out.
Evaluate all exponential expressions.
Do all multiplications and divisions in order from left to
right.
Do all additions and subtractions in order from left to
right.
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide R-20
Examples
a) 4(9  6)3  18 = 4 (3)3  18
= 4(27)  18
= 108  18
= 90
b) 15  (7  2)  20 15  5  20

3
2
3 2
27  4
3  20 23


31
31
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide R-21
R.3
Addition, Subtraction, and
Multiplication of Polynomials


Identify the terms, coefficients, and degree of a
polynomial.
Add, subtract, and multiply polynomials.
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Polynomials
Polynomials are a type of algebraic expression.
Examples:
5y  6t
1
7x  x  4
2
3
k9  7
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide R-23
Polynomials in One Variable
A polynomial in one variable is any expression of the
type
n
n 1
2
an x  an 1x
 ...  a2 x  a1x  a0 ,
where n is a nonnegative integer, an,…, a0 are real
numbers, called coefficients, and an  0. The parts of
the polynomial separated by plus signs are called
terms. The degree of the polynomial is n, the leading
coefficient is an, and the constant term is a0. The
polynomial is said to be written in descending order,
because the exponents decrease from left to right.
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide R-24
Examples
Identify the terms.
4x7  3x5 + 2x2  9
The terms are: 4x7, 3x5, 2x2, and 9.
Find the degree.
a) 7x5  3
b) x2 + 3x + 4x3
c) 5
5
3
0
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide R-25
Addition and Subtraction

If two terms of an expression have the same variables
raised to the same powers, they are called like terms,
or similar terms.
Like Terms
Unlike Terms
3y2 + 7y2
8c + 5b
4x3  2x3
9w  3y
We add or subtract polynomials by combining like
terms.
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide R-26
Examples
Add:
(4x4 + 3x2  x) + (3x4  5x2 + 7)
(4x4 + 3x4) + (3x2  5x2)  x + 7
(4 + 3)x4 + (3  5)x2  x + 7
x4  2x2  x + 7
Subtract:
8x3y2  5xy  (4x3y2 + 2xy)
8x3y2  5xy  4x3y2  2xy
4x3y2  7xy
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide R-27
Multiplication

To multiply two polynomials, we multiply each term of
one by each term of the other and add the products.
Example: (3x3y  5x2y + 5y)(4y  6x2y)
3x3y(4y  6x2y)  5x2y(4y  6x2y) + 5y(4y  6x2y)
12x3y2  18x5y2  20x2y2 + 30x4y2 + 20y2  30x2y2
18x5y2 + 30x4y2 + 12x3y2  50x2y2 + 20y2
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide R-28
More Examples
Multiply: (5x  1)(2x + 5)
= 10x2 + 25x  2x  5
= 10x2 + 23x  5
Special Products of Binomials
Multiply: (6x  1)2
= (6x)2 + 2• 6x • 1 + (1)2
= 36x2  12x + 1
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide R-29
R.4
Factoring





Factor polynomials by removing a common factor.
Factor polynomials by grouping.
Factor trinomials of the type x2 + bx + c.
Factor trinomials of the type ax2 + bx + c, a  1,
using the FOIL method and the grouping method.
Factor special products of polynomials.
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Terms with Common Factors

When factoring, we should always look first to factor out
a factor that is common to all the terms.
Example: 18 + 12x  6x2
= 6 • 3 + 6 • 2x  6 • x2
= 6(3 + 2x  x2)
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide R-31
Factoring by Grouping

In some polynomials, pairs of terms have a common
binomial factor that can be removed in the process
called factoring by grouping.
Example: x3 + 5x2  10x  50
= (x3 + 5x2) + (10x  50)
= x2(x + 5)  10(x + 5)
= (x2  10)(x + 5)
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide R-32
Trinomials of the Type x2 + bx + c
Factor: x2 + 9x + 14.
Solution:
1. Look for a common factor.
2. Find the factors of 14, whose sum is 9.
Pairs of Factors
Sum
1, 14
15
2, 7
9
The numbers we need.
3. The factorization is (x + 2)(x + 7).
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide R-33
Another Example
Factor: 2y2  20y + 48.
1. First, we look for a common factor.
2(y2  10y + 24)
2. Look for two numbers whose product is 24 and whose
sum is 10.
Pairs
Sum
Pairs
Sum
1, 24
25
2, 12
14
3, 8
11
4, 6
10
3. Complete the factorization: 2(y  4)(y  6).
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide R-34
Trinomials of the Type
ax2 + bx + c, a  1
Method 1: Using FOIL
1. Factor out the largest common factor.
2. Find two First terms whose product is ax2.
3. Find two Last terms whose product is c.
4. Repeat steps (2) and (3) until a combination is found
for which the sum of the Outside and Inside products
is bx.
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide R-35
Example
Factor: 8x2 + 10x + 3.
(8x + )(x + )
(8x + 1)(x + 3) middle terms are wrong 24x + x = 25x
(4x + )(2x + )
(4x + 1)(2x + 3) middle terms are wrong 12x + 2x = 14x
(4x + 3)(2x + 1) Correct! 4x + 6x = 10x
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide R-36
Grouping Method
1. Factor out the largest common factor.
2. Multiply the leading coefficient a and the
constant c.
3. Try to factor the product ac so that the
sum of the factors is b.
4. Split the middle term. That is, write it as a
sum using the factors found in step (3).
5. Factor by grouping.
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide R-37
Example
Factor: 12a3  4a2  16a.
1. Factor out the largest common factor, 4a.
4a(3a2  a  4)
2. Multiply a and c: (3)(4) = 12.
3. Try to factor 12 so that the sum of the factors is the coefficient
the middle term, 1.
(3)(4) = 12 and 3 + (4) = 1
4. Split the middle term using the numbers found in (3).
3a2 + 3a  4a  4
5. Factor by grouping. 3a2 + 3a  4a  4 = (3a2 + 3a) + (4a  4)
= 3a(a + 1)  4(a + 1)
= (3a  4)(a + 1)
Be sure to include the common factor to get the complete
factorization.
4a(3a  4)(a + 1)
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide R-38
of
Special Factorizations
Difference of Squares
A2  B2 = (A + B)(A  B)
Example: x2  25 = (x + 5)(x  5)
Squares of Binomials
A2 + 2AB + B2 = (A + B)2
A2  2AB + B2 = (A  B)2
Example: x2 + 12x + 36 = (x + 6)2
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide R-39
More Factorizations
Sum or Difference of Cubes
A3 + B3 = (A + B)(A2  AB + B2)
A3  B3 = (A  B)(A2 + AB + B2)
Example: 8y3 + 125 = (2y)3 + (5)3
= (2y + 5)(4y2  10y + 25)
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide R-40
R.5
Rational Expressions




Determine the domain of a rational expression.
Simplify rational expressions.
Multiply, divide, add, and subtract rational expressions.
Simplify complex rational expressions.
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Domain of Rational Expressions
The domain of an algebraic expression is the set of all real
numbers for which the expression is defined.
x2  9
Example: Find the domain of 2
.
x  3x  4
Solution: To determine the domain, we factor the denominator.
x2 + 3x  4 = (x + 4)(x  1) and set each equal to zero.
x+4=0
x1=0
x = 4
x=1
The domain is the set of all real numbers except 4 and 1.
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide R-42
Simplifying, Multiplying, and Dividing
Rational Expressions
2
x
1
Simplify:
2 x2  x  1
2
2
9
a

6
ab

3
b
Simplify:
12a 2  12b2
Solution:
Solution:
( x  1) ( x  1)
9a 2  6ab  3b 2 3(3a 2  2ab  b 2 )

2
2
12a  12b
12(a 2  b 2 )
3(3a  b)(a  b)

12( a  b)(a  b)
(2 x  1) ( x  1)
3 (3a  b) (a  b)
x2  1
( x  1)( x  1)

2 x 2  x  1 (2 x  1)( x  1)

x 1

2x  1


Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
12 4 (a  b) (a  b)
3a  b
4( a  b)
Slide R-43
Another Example
x2
x2  4
 2
x  3 x  3x  2
Multiply:
Solution:
x2
x2  4
x  2 ( x  2)( x  2)
 2


x  3 x  3 x  2 x  3 ( x  2)( x  1)

( x  2) ( x  2)( x  2)
( x  3) ( x  2) ( x  1)
( x  2)( x  2)

( x  3)( x  1)
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide R-44
Adding and Subtracting Rational
Expressions

When rational expressions have the same denominator,
we can add or subtract the numerators and retain the
common denominator. If the denominators are different,
we must find equivalent rational expressions that have
a common denominator.

To find the least common denominator of rational
expressions, factor each denominator and form the
product that uses each factor the greatest number of
times it occurs in any factorization.
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide R-45
Example
9x  2
7

3x 2  2 x  8 3x 2  x  4
Add:
Solution:
9x  2
7

(3 x  4)( x  2) (3 x  4)( x  1)
The LCD is (3x + 4)(x  1)(x  2).
9x  2
( x  1)
7
( x  2)



(3 x  4)( x  2) ( x  1) (3 x  4)( x  1) ( x  2)
9 x2  7 x  2
7 x  14


(3 x  4)( x  2)( x  1) (3x  4)( x  2)( x  1)
(3 x  4) (3 x  4)
9 x 2  16


(3 x  4)( x  2)( x  1) (3 x  4) ( x  2)( x  1)

3x  4
( x  2)( x  1)
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide R-46
Complex Rational Expressions

A complex rational expression has rational expressions
in its numerator or its denominator or both.

To simplify a complex rational expression:
 Method 1. Find the LCD of all the denominators
within the complex rational expression. Then multiply
by 1 using the LCD as the numerator and the
denominator of the expression for 1.
 Method 2. First add or subtract, if necessary, to get a
single rational expression in the numerator and in
the denominator. Then divide by multiplying by the
reciprocal of the denominator.
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide R-47
Example: Method 1
Simplify:
1

x
1

2
x
1
y
1
y2
The LCD of the four expressions is x2y2.
1

x
1

x2
1
x2 y 2
y

1 x2 y 2
y2
1
x
 
 1
 x2 

1 2 2
(x y )

y
xy 2  x 2 y xy ( y  x )
 2
 2
2
y x
(x  y2 )
1  2 2
(x y )
2 
y 
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide R-48
Example: Method 2
Simplify:
1

x
1

2
x
1
y
1
y2
1

x
1

2
x
1
1 y 1 x
  
y
x y y x

1
1 y 2 1 x2
 2 2 2
2
2
y
x
y
y x
y
x
yx

xy xy
xy
 2

y
x2
y 2  x2
 2 2
2 2
x y
x y
x2 y 2
yx
x2 y 2
yx x2 y 2
xy ( y  x)

 2



xy
y  x2
y 2  x2
x y y 2  x2
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide R-49
R.6
Radical Notation and Rational
Exponents




Simplify radical expressions.
Rationalize denominators or numerators in
rational expressions.
Convert between exponential and radical
notation.
Simplify expressions with rational exponents.
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Notation


A number c is said to be a square root of a if
c2 = a.
nth Root
A number c is said to be an nth root of a if cn = a.
The symbol n a denotes the nth root of a. The symbol
is called a radical. The number n is called the index.
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide R-51
Examples
Simplify each of the following:
a) 49 = 7, because 72 = 49.
b)  49 = 7, because 72 = 49 and 


49  (7)  7.
3
3
27
3
3
3
27 .


 because
c) 3


 
3
125 5
 5  5 125
d)
3
64  4 because (4)3 = 64.
e)
4
25 is not a real number.
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide R-52
Properties of Radicals
Let a and b be any real numbers or expression for
which the given roots exist. For any natural numbers m
and n (n  1):
1. If n is even, n a n  a
2. If n is odd, n a n  a
3. n a  n b  n ab
4. a n a
n
b
5.
n

a 
m
n
b
 a
n
(b  0)
m
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide R-53
Examples
(7)  7  7
2
a)
f)
g)
(7)3  7
b)
3
c)
4
72  36  2  6 2
e)
81

3
27

5
 35  243
245 x 6 y 5  49  5  x 6  y 4  y
 7 x3 y 2
4
d)
27 
5
 49 x 6 y 4 5 y
3  7  21
4
3
h)
5 y  7 y 2 x3
5y
y
y2
y2


25
25 5
81

 9 3
9
9
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide R-54
Another Example
Perform the operation.
3
2 3

  2
2 3 3 3
2
9 6  6 3
 3
2
 3  2  (9  1) 6  3  3
 68 6 9
 3  8 6
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide R-55
The Pythagorean Theorem
The sum of the squares of
the lengths of the legs of a
right triangle is equal to the
square of the length of the
hypotenuse:
a2 + b2 = c2.
c
a
b
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide R-56
Example
Juanita paddled her canoe across a river 525 feet wide.
A strong current carried her canoe 810 feet downstream
as she paddled. Find the distance Juanita actually
paddled, to the nearest foot.
Solution:
810 ft
c2  a 2  b2
c  5252  8102
525 ft
x
c  275,625  656,100
c  931,725
c  965.3 ft
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide R-57
Rationalizing Denominators or
Numerators
Rationalizing the denominator (or numerator) is done by
multiplying by 1 in such a way as to obtain a perfect
nth power.
Example: Rationalize the denominator.
6
6 7
42
42

 

7
7 7
7
49
Example: Rationalize
the numerator.
a b
a b a b


4
4
a b
a  b


2

2
4 a 4 b
a b
4 a 4 b
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide R-58
Rational Exponents
For any real number a and any natural numbers m and n
for which n a exists,
1/ n
a
a
a
m/n
na
 a 
m / n
n

m
 a
n
m
, and
1
a
m/n
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide R-59
Examples
Convert to radical notation and, if possible, simplify.
a) 113/ 4  4 113
b) 91/ 2  1  1  1
1/ 2
9
3
9
c)  27 
4/3
 3 (27) 4  3 531441  81, or
 27 
4/3


3
27

4
 (3) 4  81
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide R-60
More Examples
Convert each to exponential notation.
6
a) 5 8ab   8ab 6 / 5


b) 12 x4  x4 /12  x1/ 3
Simplify.
a) x7 / 8  x3/ 4  x7 / 83/ 4  x13/ 8  8 x13  8 x8  8 x5  x 8 x5
b) ( x  2)7 / 3 ( x  2)1/ 3  ( x  2)7 / 31/ 3  ( x  2)2
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide R-61
R.7
The Basics of Equation Solving


Solve linear equations.
Solve quadratic
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Linear and Quadratic Equations

A linear equation in one variable is an equation that is
equivalent to one of the form ax + b = 0, where a and b
are real numbers and a  0.

A quadratic equation is an equation that is equivalent
to one of the form ax2 + bx + c = 0, where a, b, and c
are real numbers and a  0.
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide R-63
Equation Solving Principles

For any real numbers a, b, and c,
 The Addition Principle:
If a = b is true, then a + c = b + c is true.
 The Multiplication Principle:
If a = b is true, then ac = bc is true.
 The Principle of Zero Products:
If ab = 0 is true, then a = 0 or b = 0, and if a = 0 or
b = 0, then ab = 0.
 The Principle of Square Roots:
If x2 = k, then x  k or x   k .
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide R-64
Example: Solve 2x + 5 = 7  4(x  2)

2 x  5  7  4( x  2)
2x  5  7  4x  8
Using the distributive property.
2 x  5  15  4 x
6 x  5  15
Combine like terms.
Using the addition principle.
6 x  10
Using the addition principle.
10
x
6
5
x
3
Using the multiplication principle.
Simplifying.
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide R-65
Check:

We check the result
in the original equation.
2 x  5  7  4( x  2)
5
5

2   5  7  4  2
3
3

10
5 6
 5  7  4  
3
3 3
10 15
 
3
3
25

3
25

3
21
 1
 4 
3
 3
21 4

3 3
25
3
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide R-66
Example: Solve x2  3x = 10

Write the equation with 0 on one side.
x 2  3x  10
x  3x  10  0
2
( x  2)( x  5)  0
x20
x  2
or
or
x50
x5
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide R-67
Check: x2  3x = 10

For x = 2

For x = 5
x  3 x  10
x 2  3x  10
(2)  3(2)  10
4  6  10
(5) 2  3(5)  10
25  15  10
10  10
10  10
2
2
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide R-68
Example: Solve 5x2  25 = 0

We will use the principle of square roots.
5 x 2  25  0
5 x 2  25
x 5
2
x  5 or x   5
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide R-69