Transcript Section 1.4 Powerpoint

Copyright 2013, 2009, 2005, 2002 Pearson, Education, Inc.
1.4
Properties of Real
Numbers and
Algebraic
Expressions
Algebraic Equations
Algebraic equation is a statement that
two expressions have equal value.
Equality is denoted by the phrases
•
•
•
•
•
•
•
equals
gives
is/was/should be
yields
amounts to
represents
is the same as
Example
Write each sentence as an equation.
The difference of 7 and a number is 42.
7 – x = 42
The quotient of y and twice x is the same as the
product of 4 and z
y
 4z
2x
Equality and Inequality Symbols
Symbol
a=b
ab
a<b
a>b
ab
ab
Meaning
a is equal to b.
a is not equal to b.
a is less than b.
a is greater than b.
a is less then or equal to b.
a is greater than or equal to b.
Example
Tell whether each mathematical statement is true or
false.
a.
4<5
True
b.
27 ≥ 27 True
c.
0>5
d.
16 ≤ 9
False
False
Example
Insert <, >, or = between the pairs of numbers
to form true statements.
a.
4.7 > 4.697
b.
32.61 = 32.61
c.
– 4 > –7
d.
1
4
<
2
3
Example
Translate each sentence into a mathematical statement.
a.
Thirteen is less than or equal to nineteen.
13
≤
19
b.
Five is greater than two.
5
>
2
c.
Seven is not equal to eight.
7
≠
8
Identities
Addition
0 is the identity since a + 0 = a and 0 + a = a.
Multiplication
1 is the identity since a · 1 = a and 1 · a = a.
Inverses
Additive and Multiplicative Inverses
The numbers a and –a are additive inverses or
opposites of each other because their sum is
0; that is a + (–a) = 0.
1
The numbers b and (for b ≠0) are
b
reciprocals or multiplicative inverses of each
other because their product is 1; that is
1
b  1
b
Example
Write the additive inverse, or opposite, of each.
the opposite is  5
a. 5
b.
4
5
c. 3.7
4
the opposite is 
5
the opposite is 3.7
Example
Write the multiplicative inverse, or reciprocal, of
each.
1
the reciprocal is
a. 5
5
b.
4 the reciprocal is 5
4
5
c. 3
1
the reciprocal is 
3
Commutative and Associative
Property
Commutative property
• Addition: a + b = b + a
•Multiplication: a · b = b · a
Associative property
• Addition: (a + b) + c = a + (b + c)
• Multiplication: (a · b) · c = a · (b · c)
Example
Use the commutative or associative property to
complete.
a. x + 8 = ______
8+x
b. 7 · x = ______
x·7
c. 3 + (8 + 1) = _________
(3 + 8) + 1
d. (‒5 ·4) · 2 = _________
‒5(4 · 2)
e. (xy) ·18 = ___________
x · (y ·18)
Distributive Property
For real numbers, a, b, and c.
a(b + c) = ab + ac
Also,
a(b  c) = ab  ac
Example
Use the distributive property to remove the parentheses.
7(4 + 2) =7(4 + 2) = (7)(4) + (7)(2)
= 28 + 14
=
42
Example
Use the distributive property to write each expression
without parentheses. Then simplify the result.
a. 7(x + 4y) = 7x + 28y
b. 3(‒5 + 9z) = 3(‒5) + (3)(9z) = ‒15 + 27z
c. ‒(8 + x ‒ w) = (‒1)(8) + (‒1)(x) ‒ (‒1)(w)
= ‒8 ‒ x + w
Example
Write each as an algebraic expression.
a. A vending machine contains x quarters. Write
an expression for the value of the quarters.
0.25x
b. The cost of y tables if each tables costs $230.
230y
Example
Write each as an algebraic expression.
a. Two numbers have a sum of 40. If one number
is a, represent the other number as an
expression in a.
40 – a
b. Two angles are supplementary if the sum of their
measures is 180 degrees. If the measure of one
angle is x degrees represent the other angle as
an expression in x.
180 – y
Like Terms
Terms of an expression are the addends of the
expression.
Like terms contain the same variables raised to
the same powers.
Example
Simplify by combining like terms.
a. 3x  5x  7  (3  5) x  7
 2 x  7
b. 8x  10 x  8  3  (8  10) x  ( 8  3)
 18 x  11
Example
Simplify each expression.
a. 7 x  5  3x  2  7 x  3x  5  2
 ( 7  3) x  (5  2)
 4 x  3
b. 3 y  2 y  5  7  y  3 y  2 y  5  7  y
 (3  2  1) y  ( 5  7)
 2 y  12
Example
Simplify by using the distributive property to
multiply and then combining like terms.
7 x 2  3  5( x 2  4)  7 x 2  3  5 x 2  20
 2 x 2  23
Example
Simplify by using the distributive property to
multiply and then combining like terms.
1
1
1
(4a  6b)  (9a  12b  1) 
2
3
4
1 1
 2a  3b  3a  4b  
3 4
1 1
 (2a  3a )  ( 3b  4b)  
3 4
7
  a  7b 
12