Download Open Sentences A mathematical statement consisting of two

Open Sentences
A mathematical statement consisting of two expressions with one or more variables.
I.
Equations
An open sentence that contain two expressions which are equated.
Expression 1 = Expression 2
A.
Solve Equations
1.
Using a Replacement Set
Substitute each element in the replacement set for the specified variable into the equation and simplify. Those elements that
produce a true statement are members of the solution set.
Given: replacement set: {2, 3, 4, 5, 6}
Find solution set for 4a + 7 = 23
4(2) + 7 = 23
4(3) + 7 = 23
4(4) + 7 = 23
4(5) + 7 = 23
4(6) + 7 = 23
8 + 5 = 23
12 + 7 = 23
16 + 7 = 23
20 + 7 = 23
24 + 7 = 23
13 = 23
19 = 23
23 = 23
27 = 23
31 = 23
False
False
True
False
True
Solution Set: {4}
2.
Using Order
of Operations
Solve k = [5(8 + 2)] / [18 - (5 - 3)3]
k = [5(8 + 2)] / [18 - (5 - 3)3]
k = [5(10)] / [18 - (2)3]
k = [5(10)] / [18 - 8]
k = [50] / [18 - 8]
k = [50] / [10]
k= 5
II.
Inequalities
An open sentence that contains two expressions which are related by an inequality.
Expression 1 (>, >=, <, <=) Expression 2
A.
Solve Inequalities
1.
Using a Replacement Set
Substitute each element in the replacement set for the specified variable into the inequality and simplify. Those elements that
produce a true statement are members of the solution set.
Given: replacement set {20, 21, 22, 23}
Find solution set for z + 11 > 32
20 + 11 > 32
21 + 11 > 32
22 + 11 > 32
23 + 11 > 32
31 > 32
32 > 32
33 > 32
34 > 32
False
False
True
True
Solution Set: {22, 23}
2.
Using an Estimate
This is essentially a 'trial and error' method using an 'educated' guess.
Solve 20 + 4z > 80 using estimation, z is an integer.
first estimate = 10
20 + 4(10) > 80
20 + 40 > 80
60 > 80
False, value too small, difference is 20, divide by 4, increase by 5
second estimate = 15
20 + 4(15) > 80
20 + 60 > 80
80 > 80
False, value too small, difference is 0, increase by 1
third estimate = 16
20 + 4(16) > 80
20 + 64 > 80
84 > 80
True, solution set included all integers greater than or equal to 16
Solution Set: {z | z >= 16}
III. Expressions
An expression consists of one or more numbers and variables with one or more arithmetic operators.
Expressions can be written
• as an algebraic expression using symbols,
• as a verbal expression using words.
A.
Algebraic Expression
An expression that uses symbols and consists of one or more numbers and variables with one or more arithmetic operators.
Examples
4a
6a + 2
8 - 9w
3ac / 4ab2
1.
Terms
A term is a number, a variable, or a product or quotient of numbers and variables.
7, 4z, d,
a.
dz,
d2,
8z2d
Coefficients
The coefficient of a term is the numerical factor in a term.
ex.
in 4vm, 4 is the coefficient
in y2, 1 is implied
b.
Like Terms
Like terms are terms that contain the same variables with corresponding variables having the same power.
ex.
4x2 + 3xc - 2xc + 3c,
3xc and 2xc are like terms
2.
Equivalent Expressions
Equivalent expressions represent the same value.
ex.
10x + 6x
and
16x
10x + 6x = 16x
x(10 + 6) = 16x
x(16) = 16x
16x = 16x
3.
Simplest Form
An expression is in simplest form
when it is replaced by an equivalent expression that has no like terms or parentheses.
B.
Verbal Expression
An expression that uses words and consists of one or more numbers and variables with one or more arithmetic operators.
Examples
• sum of four and six
• difference of nine and two
• product of six and five
• quotient of eight and two.
1.
Translation Examples
algebraic to
verbal
4z + 3
the sum of four times z and three.
9x4 - 8
the difference of nine times x to the fourth power and eight.
verbal to algebraic
3 / x3
the quotient of three and x cubed
five times the difference of twelve and y
C.
5(12 - y)
Order of Operations
Step 1. Evaluate expressions inside grouping symbols (from the inside out)
Step 2. Evaluate all powers
Step 3. Do all multiplication and/or divisions from left to right
Step 4. Do all additions and/or subtractions from left to right
1.
PEMDAS
Please -
parenthesis
-------------------------------------Excuse - exponents
-------------------------------------My -
multiplication
Dear -
division
-------------------------------------Aunt -
addition
Sally -
subtraction
a.
Numeric Example (worked)
6 + 23
6+8
14
-------- = -------- = --- = 2
32 - 2
b.
9-2
7
Numeric Example
(8 - 3) * 3(3 + 2)
5 * 3(5)
5 * 15
75
c.
Numeric Example
25 - 6 * 2
64 - 6 * 2
64 - 12
52
52
----------------- = ---------------- = --------------- = --------- = ----- = 5.2
33 - 5 * 3 - 2
9-5*3-2
27 - 15 - 2
12 - 2
10
d.
Algebraic Example
w / substitutions
evaluate 2(x2 - y) + z2
if x = 4, y = 3, and z = 2
2(42 - 3) + 22
2(16 - 3) + 4
2(13) + 4
26 + 4
30
D.
Numbers
A number from the real number set
• whole numbers
• natural numbers
• integers
• decimals
• fractions
E.
Variables
A symbol (often letters) that is used to represent some unspecified number or value.
F.
Operators
Addition: Two or more numbers added together is called a sum.
9 + 6 = 15
Subtraction: One number subtracted form another number is called a difference.
10 - 3 = 7
Multiplication: One number (factor) multiplied (times) another number (factor) is called a product.
5 * 3 = 15
Division: One number (dividend) divided by another number (divisor) is called a quotient.
6/3=2
1.
Powers
An expression in the form of xn, read "x to the n th power".
x is called the base
n is called the exponent
The exponent represents the number of times the base is used as a factor.
four to the third power
43 = 4 * 4 * 4 = 64
IV. Algebraic Properties of Equality
The basic rules of equality in Algebra
A.
Distributive Property
For any number a, b, and c,
a(b + c) = ab + ac and (b + c)a = ba + ca
and
a(b - c) = ab - ac and (b - c)a = ba - ca
1.
Distribute Over Addition
For any number a, b, and c,
a(b + c) = ab + ac
2(4 + 5) = 2(4) + 2(5)
2(9) = 8 + 10
18 = 18
2.
Distribute Over Subtraction
For any number a, b, and c,
a(b - c) = ab - ac and (b - c)a = ba - ca
3(6 - 2) = 3(6) - 3(2)
3(4) = 18 - 6
12 = 12
B.
Commutative Property
The order in which you add or multiply numbers does not change their sum or product.
For any numbers a and b,
a+b=b+a
(Addition)
and
a•b=b•a
ex.
4+3=3+4
5•6=6•5
(Multiplication)
C.
Associative Property
The way you group three or more numbers when adding or multiplying does not change their sum or
product.
For any numbers a, b and c,
(a + b) + c = a + (b + c)
(Addition)
and
(ab)c = a(bc)
ex.
(Multiplication)
(5 + 4) + 3 = 5 + (4 + 3)
(3 • 5) • 2 = 3 • (5 • 2)
D.
Additive Identity
Property
For any number a, the sum of a and 0 is a.
a+0=0+a=a
8+0=0+8=8
E.
Multiplicative Identity Property
For any number a, the product of a and 1 is a.
a•1=1•a=a
99 • 1 = 1 • 99 = 99
F.
Multiplicative Property of Zero
For any number a, the product of a and 0 is 0.
a•0=0•a=0
9•0=0•9=0
G. Multiplicative Inverse Property
For every number a/b, where a, b not equal to zero, there is exactly one number b/a such that the
product of a/b • b/a is 1.
a/b • b/a = b/a • a/b = 1
9/5 • 5/9 = 45/45 = 1
H. Reflexive Property
Any quantity is equal to itself.
For any number a, a = a.
5=5
I.
Symmetric Property
If one quantity equals a second quantity, then the second quantity equals the first.
For any numbers a and b, if a = b then b = a.
If 10 = 8 + 2 then 8 + 2 = 10
J.
Transitive Property
If one quantity equals a second quantity and the second quantity equals a third quantity, the the first
quantity equals the first quantity.
For any numbers a, b, and c, if a = b and b = c, then a = c.
if 2 + 7 = 8 + 1, and 8 + 1 = 5 + 4, then 5 + 4 = 2 + 7.
K. Substitution Property
A quantity may be substituted for its equal in any expression.
If a = b, then a may be replaced by b in any expression.
If r = 22, then 4r = 4 • 22.
Language of Algebra
wetb - 2009
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