Download Number Sets Properties and Solving Equations WS

NOTES – Number Systems & Properties
Algebra II Pre AP
Natural Numbers: N, “counting numbers” {1,2,3,…}
Whole Numbers: W, {0,1,2,3…}
Integers: Z, {…-2, -1, 0, 1, 2…}
Rational: Q, any number that can be expressed in
a
the form , where a and b are
b
integers and b  0
Real Numbers:  , union of Rational and Irrational
Irrational: I, nonterminating, nonrepeating decimals
Real Numbers 
Venn Diagram
Q
Z
W
I
N
Name all the sets of numbers to which each belongs.
1.) 0 _____________
2.)  81 ___________
3.) 
11
_____________
3
4.)  ____________
5.) 6.25 ___________
6.) 1.3 _____________
7.)
19 _____________
8.)
8 ___________
10.) 60 _____________
11.)
7
______________
0
12.) 4 __________
9.)
3
0
____________
5
Graph each of the following on a number line.
13.) all whole numbers less than 5
14.) all integers between -3 and 4
15.) all integers between -3 and 4 inclusive
16.) all natural numbers greater than -2
17.) all real numbers less than or equal to 4
18.) whole numbers greater than -3
19.) integers between -6 and -2 inclusive
20.) natural numbers less than 1
True or False: If false, give an example of a number that shows the statement is false.
21.) Every real number is irrational. ________
22.) Every integer is a rational number. ______
23.) Every rational number is an integer. _____
24.) Every natural number is an integer. _____
25.) Every irrational number is a real number. ______
26.) Every real number is either a rational or an irrational number. _____
Properties of Real Numbers
Property
Addition
Multiplication
Commutative
ab  ba
ab  ba
Associative
(a  b)  c  a  (b  c)
(ab)c  a(bc)
a0  a
a 1  a
1 a a
Identity
Inverse
0a  a
 a   a  0;
a   a   0
Distributive of Multiplication over Addition
1
1
  a  1; a    1
a
a
a(b  c)  ab  ac; (b  c)a  ba  ca
a 0  0
0 a0
Name the property illustrated by each equation below.
Multiplicative Property of Zero
1.) 5x   4 y  3x   5x   3x  4 y 
6.) 7 x   9 x  8   7 x  9 x   8
2.) 5  3x  y   5  3x  1y 
7.) 7n  2n   7  2  n
3.)  2 x  y  2  xy 
8.) 3x  2 y  3  2  x  y
4.)  6  (6)  y  0 y
9.)
5.) 5  x  y   5x  5 y
10.) 4n  0  4n
1
 4 y  1y
4
Write the additive inverse and the multiplicative inverse for each of the following.
3
11.) 
_____ _____ 12.) 0.6 _____ _____ 13.) 0 _____ _____ 14.) 1 _____ _____
7
Tell whether the statement is always, sometimes, or never true for real numbers a, b, and c. Explain.
15.)  a  b   c  a   b  c  __________________________________________________________
16.)
 a  b   c  a  b  c 
___________________________________________________________
17.)
 a  b   c  a  b  c 
___________________________________________________________
18.)
 a  b  c  a  b  c 
___________________________________________________________
19.)
a  b  c   ab  ac
___________________________________________________________
20.)
a  b  c   ab  ac
___________________________________________________________
Solve each equation
1.)
2
3
m m  4
3
5
2.)
4.) 5  2  x   3  2 x  7  3x
3
2 4
1
w  w
7
9 9
7
3.)
2
1 3
1
k  k
5
6 10
3
5.) 2  4  3x   7  6  x  1
6.) Solve the equation ax  b  cx  d for x in terms of a, b, c, and d.
Under what conditions is there no solution?
Under what conditions are all real numbers solutions?
Solve each equation and state restrictions if needed
7.) x  y  2   z; solve for y
8.) I  prt; solve for t
9.) F  G
Mm
; solve for M
r2
Solve for x and state restrictions if needed
10.) ax  bx  2c
11.) x  cx  d
13.) 2dx  e  x
14.)
16.) Show that
1
 a  b; when x  0
x
12.) ax  b  x
15.) b  cx  dx
a c a b
   for nonzero real numbers a, b, c, and d. Justify each step in your reasoning.
b d c d
a
c
and
be two distinct rational numbers. Find the rational number that lies exactly halfway
b
d
a
c
between
and
on a number line
b
d
17.) Let