Writing Mathematics With Style

[Pages:76]Writing Mathematics With Style

An (algebraic) expression is any combination of numbers, variables, exponents, mathematical symbols, and mathematical operations, but never include equal signs or inequality signs!

An equation is a mathematical statement that two quantities are equal. Therefore, an equation consists of two expressions separated by an equal sign.

An inequality is a mathematical statement expressing an order relationship. Therefore, an inequality consists of two (or more) expressions separated by one of the inequality signs. If more than one inequality sign are used, they must be in the same direction!

Classify the following as an expression, equation, inequality, or nonsense:

1) 2x 5 2) 2x 5 = 1 3) y ? z 9 4) y ? z + 9

12 5) x = y

12 6) x y 7) 2 x 3 8) 2 x + 3

9) | 2x ? 3 | > 1 10) a < b < c 11) 2 ? 3(4 + x)2 ? 3x(x ? 3) 12) 2 ? 3(4 + x)2 = 3x(x ? 3) 13) x2 = x + 6 14) x2 x + 6 15) x2 x + 6 16) x2 x 6 17) x2 x 6

Simplifying Expressions:

You usually simplify expressions by performing a series of mathematical operations to obtain another expression. To express the fact that the new expression is equal to the old expression, use the equal sign (=). Always arrange problems vertically, if possible. In the examples below, start with the given expression, and simplify to the answer.

Example 1: Simplify: GivenExpression

GivenExpression = expression1 = expression2 = expression3 = answer

GivenExpression = expression1 = expression2 = expression3 = answer

Example 2: Simplify 16 + 8 ? 4 ? 2 + 4

16 + 8 ? 4 ? 2 + 4

= 16 + 2 ? 2 + 4 = 16 + 4 + 4 = 20 + 4 = 24

Example 3:

= =

Simplify (x ? y)(x + y ? 2)

(x ? y)(x + y ? 2) x2 + xy ? 2x ? xy ? y2 + 2y x2 ? 2x ? y2 + 2y

DO NOT:

1. Use bars under equations or expressions to show addition or subtraction.

2. Place extraneous calculations or comments in the body of a simplification or a solution.

Solving Equations and Inequalities:

You usually solve equations and inequalities by performing a series of mathematical operation to obtain another equation or inequality. To express the fact that the new equation is implied by the old equation, sometimes we use the implication sign ( ), although it is acceptable if problems are arranged neatly. If possible, arrange problems vertically. In the examples below, start with the given equation, and solve.

Example 1: Simplify: GivenEquation

Either form given below is acceptable.

GivenEquation equation1 equation2 equation3 answer

GivenEquation equation1 equation2 equation3 answer

Example 2: Solve 2x + 1 = 3x ? 5

2x + 1 = 3x ? 5 2x + 1 ? 3x ? 1 = 3x ? 5 ? 3x ? 1

?x = ?6 x= 6

Example 3: Solve 2x + 1 3x ? 5

2x + 1 3x ? 5 2x + 1 ? 3x ? 1 3x ? 5 ? 3x ? 1

?x ?6 x 6

Types of Numbers

Natural Numbers (Counting Numbers) N N = {1, 2, 3, 4, 5, ...}

Whole Numbers W W = {0, 1, 2, 3, 4, 5, ...}

Integers Z Z = {..., ?4, ?3, ?2, ?1, 0, 1, 2, 3, 4, ...}

Rational Numbers Q a

Q = { b | a, b, Z , b 0}

Irrational Numbers I Numbers that can be written as an infinite nonrepeating decimal

Real Numbers R Any number that is rational or irrational (R = Q I)

Real Number Line

Visualize a line with equally spaced markers each of which is associated with the integers. If the integers have their natural order, then the real numbers can be visualized as points on the line.

-3 -2 -1 0 1 2 3

We notice that

1. Every real number corresponds to a unique point on the line. 2. Every point on the line corresponds to a unique real number.

This is why the set of real numbers is sometimes referred to as the real number line.

Absolute Value

| x |

=

x

if

x 0

?x if x < 0

The absolute value of a real number is equal to its (positive) distance to the origin!

Rule for Order of Operations (Please Excuse My Dear Aunt Sally)

P 1. arentheses: Simplify all groupings first.

E 2. xponents: Calculate exponential powers and radicals.

M D 3. ultiplication and ivision: Perform all multiplications and

divisions as they occur from left to right.

A S 4.

ddition and ubtraction: Perform all additions and

subtractions as they occur from left to right.

Evaluate the following expressions and note the use of the equal signs because we use mathematics writing style:

Ex 1: (4 ? 6)2 + 6(?4) + 5

Ex 2: 6 + 24 ? 3 ? 2 + 3 16

(4 ? 6)2 + 6(?4) + 5 (P) = (?2)2 + 6(?4) + 5 (E) = 4 + 6(?4) + 5 (M) = 4 ? 24 + 5 (S) = ?20 + 5 (A) = ?15

6 + 24 ? 3 ? 2 + 3 16 (E) = 6 + 24 ? 3 ? 2 + 3 ? 4 (D) = 6 + 8 ? 2 + 3 ? 4 (M) = 6 + 16 + 3 ? 4 (M) = 6 + 16 + 12 (A) = 22 + 12 (A) = 34

Ex 3: 2 [ 5 + 2 ( 6 + 3 ? 4) ]

2 [ 5 + 2 ( 6 + 3 ? 4) ] = 2 [ 5 + 2 ( 9 ? 4) ] = 2 [ 5 + 2 ( 5) ] = 2 [ 5 + 10 ] = 2 [ 15 ] = 30

Ex 4: 10 + 12 ? 4 + 2 ? 3

10 + 12 ? 4 + 2 ? 3 = 10 + 3 + 2 ? 3 = 10 + 3 + 6 = 13 + 6 = 19

Types of Intervals

Interval Notation

Graph

Algebraic Notation

Interval Description

(a, b)

a

b

a < x < b

Open, finite

[a, b]

a

b

a x b

Closed, finite

[a, b)

a

b

a x < b Half?open, finite

(a, b]

a

b

a < x b Half?open, finite

(a, ) a

x > a

Open, infinite

(?, b)

x < b

Open, infinite

b

[a, ) a

x a

Closed, infinite

(?, b]

x b

Closed, infinite

b

If your answer is composed of two (or more) distinct intervals, then the algebraic form of your answer must contains the conjunction 'OR'.

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