Transcript Blair, Tobey, Slater

Chapter 1
Real
Numbers &
Variables
© 2010 Pearson Prentice Hall. All rights reserved.
§ 1.1
Adding Real Numbers
© 2010 Pearson Prentice Hall. All rights reserved.
Different Types of Numbers
Whole numbers are numbers such as 0, 1, 2, 3, 4, 5, ….
Integers are numbers such as …, 3, 2, 1, 0, 1, 2, 3, …
Rational Numbers are numbers such as  23 , 45 ,  87 , 12 , and
8.
9
Irrational Numbers are numbers that cannot be expressed
as one integer divided by another integer. The numbers π,
2, and 8 are irrational numbers.
5
Real Numbers are all rational and irrational numbers.
© 2010 Pearson Prentice Hall. All rights reserved.
Tobey & Slater, Beginning Algebra, 7e
3
The Number Line
A number line is a line on which each point is
associated with a number.
–5 –4 –3 –2 –1
0
– 4.8
Negative
numbers
1
2
3
4
5
1.5
Positive
numbers
The real numbers include the positive numbers,
the negative numbers, and zero.
© 2010 Pearson Prentice Hall. All rights reserved.
Tobey & Slater, Beginning Algebra, 7e
4
Opposite Numbers
Opposite numbers (or additive inverses) have the
same magnitude but different signs.
The opposite of 4 is – 4.
–5 –4 –3 –2 –1
0
1
4 + (– 4) = 0
10 – 4 = 6
10 + (–4) = 6
15 – 8 = 7
15 + (–8) = 7
12 – 2 = 10
12 + (–2) = 10
© 2010 Pearson Prentice Hall. All rights reserved.
2
3
4
5
The sum of a number and
its opposite is zero.
Subtracting is the same
as adding the opposite.
Tobey & Slater, Beginning Algebra, 7e
5
Absolute Value
The absolute value of a number is the distance
between that number and zero on a number line.
| – 4| = 4
Symbol for
absolute
value
|5| = 5
Distance of 4
–5 –4 –3 –2 –1
Distance of 5
0
© 2010 Pearson Prentice Hall. All rights reserved.
1
2
3
4
5
Tobey & Slater, Beginning Algebra, 7e
6
Adding Two Numbers with Same Signs
Addition Rule for Two Numbers With the Same Sign
To add two numbers with the same sign, add the absolute
values of the numbers and use the common sign in the
answer.
Example:
Add (– 3) + (– 11).
–3
+ –11
–14
Add the absolute values of
the numbers 3 and 11.
A negative sign is used because we added two negative numbers.
© 2010 Pearson Prentice Hall. All rights reserved.
Tobey & Slater, Beginning Algebra, 7e
7
Adding Two Numbers with Different Signs
Addition Rule for Two Numbers With Different Signs
1. Find the difference between the larger absolute value
and the smaller one.
2. Give the answer the sign of the number having the
larger absolute value.
Example:
Add 5 + (– 9).
5
+ –9
–4
Subtract the absolute values of the
numbers 5 and 9.
A negative sign is used because the sign of the larger number is negative.
© 2010 Pearson Prentice Hall. All rights reserved.
Tobey & Slater, Beginning Algebra, 7e
8
Adding Two Numbers with Different Signs
Example:
Add (–24) + (38).
– 24
+ 38
14
Subtract the absolute values of the
numbers 24 and 38.
The answer is positive because the sign of the larger number is positive.
Example:
Add (– 36) + 4.
– 36
+ 4
– 32
© 2010 Pearson Prentice Hall. All rights reserved.
Tobey & Slater, Beginning Algebra, 7e
9
Addition Properties for Real Numbers
1. Addition is commutative.
If two numbers are added, the result is the same no matter
which number is written first. The order of the numbers does
not affect the result.
3 + (1) = (1) + 3 = 2
2. Addition of zero to any number will result in that given
number.
4+0=0+4=4
3. Addition is associative.
If three numbers are added, it does not matter which two
numbers are grouped together and added first.
2 + (3 + 1) = (2 + 3) + 1
2+4=5+1
6=6
© 2010 Pearson Prentice Hall. All rights reserved.
Tobey & Slater, Beginning Algebra, 7e
10
Addition Properties for Real Numbers
Example:
Add (–56) + 6 + (–14).
Because addition is commutative, the numbers can be added
in any way.
(–56) + 6 + (–14)
– 50
+ (–14)
– 64
© 2010 Pearson Prentice Hall. All rights reserved.
(–56) + (–14) + 6
or
– 70
+6
– 64
Tobey & Slater, Beginning Algebra, 7e
11
§ 1.2
Subtracting Real
Numbers
© 2010 Pearson Prentice Hall. All rights reserved.
Subtracting Real Numbers
Subtraction of Real Numbers
To subtract real numbers, add the opposite of the second
number (the number you are subtracting) to the first.
Example:
Subtract. –6 – 14
–6 + (–14)
The opposite of 14 is –14.
Change the subtraction to addition.
–20
Perform the addition of the two
negative numbers.
© 2010 Pearson Prentice Hall. All rights reserved.
Tobey & Slater, Beginning Algebra, 7e
13
Subtracting Real Numbers
Example:
Subtract. –21 – (–13)
The opposite of –13 is 13.
–21 + (13)
Change the subtraction to addition.
–8
Perform the addition.
© 2010 Pearson Prentice Hall. All rights reserved.
Tobey & Slater, Beginning Algebra, 7e
14
Subtracting Real Numbers
Example:
Subtract. 5  1
11 5
5 1 5  1
Subtraction changed to addition.
    
11 5 11  5 
5 5
1 11
        The LCD is 55.
11 5   5 11
25  11  Multiply the fractions.

 
55  55 
14
Add the fractions.

55
© 2010 Pearson Prentice Hall. All rights reserved.
Tobey & Slater, Beginning Algebra, 7e
15
§ 1.3
Multiplying & Dividing
Real Numbers
© 2010 Pearson Prentice Hall. All rights reserved.
Multiplication with Same Signs
This number
decreases by
1 each time.
3(–4) = –12
2(–4) = –8
1(–4) = –4
0(–4) =
–1(–4) =
–2(–4) =
0
4
8
–3(–4) =
12
This number
increases by
4 each time.
We can see the general rule that when two numbers with the
same sign are multiplied, the result is positive.
© 2010 Pearson Prentice Hall. All rights reserved.
Tobey & Slater, Beginning Algebra, 7e
17
Multiplication with Different Signs
This number
decreases by
1 each time.
3(4)
2(4)
1(4)
0(4)
–1(4)
–2(4)
–3(4)
=
=
=
=
=
=
=
12
8
4
0
–4
–8
–12
This number
decreases by
4 each time.
We can see the general rule that when two numbers with
opposite signs are multiplied, the result is negative.
© 2010 Pearson Prentice Hall. All rights reserved.
Tobey & Slater, Beginning Algebra, 7e
18
Multiplication with the Same Signs
Multiplication of Real Numbers
To multiply two real numbers with the same sign,
multiply the absolute values. The sign of the result is
positive.
Example:
Multiply. –75 × (– 3)
–75 × (– 3) = 225
Example:
2
Multiply.  5 
 

 12  3 
The result will always be positive.
1
 5  2   5
 12  3  18
 6  
© 2010 Pearson Prentice Hall. All rights reserved.
Tobey & Slater, Beginning Algebra, 7e
19
Multiplication with Opposite Signs
Multiplication of Real Numbers
To multiply two real numbers with opposite signs,
multiply the absolute values. The sign of the result is
negative.
Example:
Multiply. –6(4)
–6(4) = –24
Example:
Multiply. 12(–9)
The result will always be negative.
12(–9) = –108
© 2010 Pearson Prentice Hall. All rights reserved.
Tobey & Slater, Beginning Algebra, 7e
20
Division with the Same Signs
Division of Real Numbers
To divide two real numbers with the same sign, divide
the absolute values. The sign of the result is positive.
Example:
Divide. –75 ÷ (– 3)
–75 ÷ (– 3) = 25
Example:
Divide.  5    3 
 12   2 
The result will always be positive.
1
 5  2   5
 12  3  18
 6  
© 2010 Pearson Prentice Hall. All rights reserved.
Tobey & Slater, Beginning Algebra, 7e
21
Division with Opposite Signs
Division of Real Numbers
To divide two real numbers with opposite signs, divide
the absolute values. The sign of the result is negative.
Example:
Divide. –6 ÷ 2
–6 ÷ 2 = –3
Example:
Divide. 120 ÷ (–10)
The result will always be negative.
120 ÷ (–10) = –12
© 2010 Pearson Prentice Hall. All rights reserved.
Tobey & Slater, Beginning Algebra, 7e
22
§ 1.4
Exponents
© 2010 Pearson Prentice Hall. All rights reserved.
Exponents
An exponent is a “shorthand” number that saves
writing the multiplication of the same numbers.
3 3 3 3  3
4
exponent
34
base
This is read “three to the fourth power.”
© 2010 Pearson Prentice Hall. All rights reserved.
Tobey & Slater, Beginning Algebra, 7e
24
Evaluating Numerical Expressions
In algebra, when a number is not known, a variable is
used to represent the number.
If an unknown number is represented by a, and this
number occurs as a factor three times, we can write
(a)(a)(a) = a3.
“a cubed”
(a)(a) = a2
“a squared”
© 2010 Pearson Prentice Hall. All rights reserved.
Tobey & Slater, Beginning Algebra, 7e
25
Sign Rule for Exponents
Sign Rule for Exponents
Suppose a number is written in exponent form and the
base is negative. The result is positive if the exponent is
even. The result is negative if the exponent is odd.
Example:
Evaluate (3)3.
(3)3 = (3)(3)(3) = 27
Odd exponent: Negative result
© 2010 Pearson Prentice Hall. All rights reserved.
Tobey & Slater, Beginning Algebra, 7e
26
Sign Rule for Exponents
Example:
Evaluate 26.
26 = (2)(2)(2)(2)(2)(2) = 64
Even exponent: Positive result
Example:
4
2
Evaluate    .
5
4
2
2
2
2
2
            
5
5555
16

625
© 2010 Pearson Prentice Hall. All rights reserved.
The negative sign is
NOT raised to the
fourth power!
Tobey & Slater, Beginning Algebra, 7e
27
§ 1.6
Using the Distributive
Property to Simplify
Algebraic Expressions
© 2010 Pearson Prentice Hall. All rights reserved.
Simplifying Algebraic Expressions
An algebraic expression is a quantity that contains
numbers and variables.
x + y , 3a2  a , 3x + 2y  z
Terms
A term is a number, a variable, or a product of numbers
and variables.
© 2010 Pearson Prentice Hall. All rights reserved.
Tobey & Slater, Beginning Algebra, 7e
29
Distributive Property
An important property of algebra is the distributive
property.
Distributive Property
For all real numbers a, b, and c,
a(b + c) = ab + ac
7(4 + 2) = (7)(4) + (7)(2)
7(6) = 28 + 14
42 = 42
© 2010 Pearson Prentice Hall. All rights reserved.
Tobey & Slater, Beginning Algebra, 7e
30
Distributive Property
Example:
Simplify. (5)(x  3y)
(5)(x  3y) = (5)(x)  (5)(3y)
= 5x  (15y)
= 5x + 15y
Example:
1
2
1

Simplify.  5 x  3 y  4  3
Sometimes the number may be
to the right of the parentheses.


 1 x  2 y  1  3   1 x (3)   2
5
5 
3
3
4 

 

3
3
 x  2y 
5
4
© 2010 Pearson Prentice Hall. All rights reserved.
1


y (3)    (3)

 4
Tobey & Slater, Beginning Algebra, 7e
31
Distributive Property
Example:
Simplify. 2.4(x + 3.5y) + 2y(1.4x + 1.9y)
2.4(x + 3.5y) + 2y(1.4x + 1.9y)
= 2.4(x) + 2.4(3.5y) + 2y(1.4x) + 2y(1.9y)
= 2.4x + 8.4y + 2.8xy + 3.8y2
© 2010 Pearson Prentice Hall. All rights reserved.
Tobey & Slater, Beginning Algebra, 7e
32
§ 1.7
Combining Like Terms
© 2010 Pearson Prentice Hall. All rights reserved.
Like Terms
Like terms have identical variables and identical
exponents.
20x + 5y + 2y + (3)
Like terms
To combine like terms, you add or subtract the like terms.
20x + 5y + 2y + (3) = 20x + 7y + (3)
Like terms combined.
© 2010 Pearson Prentice Hall. All rights reserved.
Tobey & Slater, Beginning Algebra, 7e
34
Combining Like Terms
Example:
Simplify the expression.
12a3 + 5b3 + a + (3a3) + 8a
Like terms
Like terms
12a3 + ( 3a3) + a + 8a + 5b3
A variable without a numerical
coefficient has a coefficient of 1.
9a3 +9a + 5b3
© 2010 Pearson Prentice Hall. All rights reserved.
Tobey & Slater, Beginning Algebra, 7e
35
Combining Like Terms
Example:
2
3 2 1
7 2
Simplify the expression. c  d  c  d
5
Like terms
Like terms
4
3
8
2
3 2 1
7 2 2
1
3 2 7 2
c d  c d  c c d  d
5
4
3
8
5
3
4
8
2 3
1 5
3 2 2 7 2
  c  c  d  d
5 3
3 5
4 2
8
Find the LCD.
6
5
6 2 7 2
 c c d  d
15
15
8
8
1
1 2
 c d
15
8
© 2010 Pearson Prentice Hall. All rights reserved.
Tobey & Slater, Beginning Algebra, 7e
36
Combining Like Terms
Example:
Simplify the expression. 8(3x – 2y) + 4(3y – 5x)
8(3x – 2y) + 4(3y – 5x)
= 24x – 16y + 12y – 20x
Distribute.
= 4x – 4y
Combine like terms.
© 2010 Pearson Prentice Hall. All rights reserved.
Tobey & Slater, Beginning Algebra, 7e
37