Archive for May, 2014

Use number lines

Use number lines 

A good way to get a picture of a set

of numbers is by using a number line. To construct a number line, choose

any point on a horizontal line and label it 0. Next, choose a point to the right

of 0 and label it 1. The distance from 0 to 1 establishes a scale that can be

used to locate more points, with positive numbers to the right of 0 and negative

numbers to the left of 0. The number 0 is neither positive nor negative.

A number line is shown in Figure 1.

figure

 

The set of numbers identified on the number line in Figure 1, including

positive and negative numbers and 0, is part of the set of integers, written

I 5 {. . . , 23, 22, 21, 0, 1, 2, 3, . . . }.

Each number on a number line is called the coordinate of the point that

it labels, while the point is the graph of the number. Figure 2 shows a number

line with several selected points graphed on it.

figure1

Write sets using set notation

Write sets using set notation 

 A set is a collection of objects called the elements or members of the set. In algebra, the elements of a set are usually numbers. Set braces, { }, are used to enclose the elements.

For example, 2 is an element of the set {1, 2, 3}. Since we can count the number of elements in the set {1, 2, 3}, it is a finite set.

 

In our study of algebra, we refer to certain sets of numbers by name. The set

N 5 {1, 2, 3, 4, 5, 6, . . . } is called the

natural numbers or the counting numbers. The three dots

show that the list continues in the same pattern indefinitely. We cannot list

all of the elements of the set of natural numbers, so it is an infinite set.

When 0 is included with the set of natural numbers, we have the set of

whole numbers, written W 5 {0, 1, 2, 3, 4, 5, 6, . . . }.

A set containing no elements, such as the set of whole numbers less than 0,

is called the empty set, or null set, usually written .

 

Caution

Do not write {ø } for the empty set; { ø } is a set with one element, . Use

only the notation for the empty set.

 

In algebra, letters called variables are often used to represent numbers

or to define sets of numbers. For example,

{x | x is a natural number between 3 and 15}

(read “the set of all elements x such that x is a natural number between 3 and

15”) defines the set

{4, 5, 6, 7, . . . , 14}.

The notation {x | x is a natural number between 3 and 15} is an example

of set-builder notation.

{ x | x has property P }

the set of all elements x such that x has a given property P

 

Simplifying Complex Rational Expressions

Simplifying Complex Rational Expressions

A complex rational expression, or complex fraction expression, is a rational

expression that has one or more rational expressions within its numerator or

denominator. Here are some examples:

 

\frac{1+\frac{2}{x}}{3}

There are two methods to simplify complex rational expressions. We will

consider them both.

 

Method 1

 

Multiplying by the LCM of all the Denominators

To simplify a complex rational expression:

a. First, find the LCM of all the denominators of all the rational

expressions occurring within both the numerator and the

denominator of the complex rational expression.

b. Then multiply by 1 using LCM LCM.

c. If possible, simplify by removing a factor of 1.

 

Method 2

Adding in the Numerator and the Denominator

To simplify a complex rational expression:

a. Add or subtract, as necessary, to get a single rational expression in

the numerator.

b. Add or subtract, as necessary, to get a single rational expression in

the denominator.

c. Divide the numerator by the denominator.

d. If possible, simplify by removing a factor of 1.

 

Subtracting Rational Expressions

Subtracting Rational Expressions

We subtract rational expressions as we do rational numbers.

To subtract when the denominators are the same, subtract the numerators and keep the same denominator. Then simplify if possible.

Subtract  \frac{8}{x}-\frac{3}{x}

\frac{8}{x}-\frac{3}{x}=\frac{8-3}{x}=\frac{5}{x}

To subtract rational expressions with different denominators:

a. Find the LCM of the denominators. This is the least common

denominator (LCD).

b. For each rational expression, find an equivalent expression with

the LCD. To do so, multiply by 1 using a symbol for 1 made up of

factors of the LCD that are missing from the original denominator.

c. Subtract the numerators. Write the difference over the LCD.

d. Simplify if possible.

 

Adding Rational Expressions

Adding Rational Expressions

Adding Rational Expression with Like Denominators

 

To add when the denominators are the same, add the numerators and keep the same denominator. Then simplify if possible.

Example

\frac{x}{x+1}+\frac{2}{x+1}+\frac{x+2}{x+1}.

 

To add rational expressions with different denominators:

a. Find the LCM of the denominators. This is the least common

denominator (LCD).

b. For each rational expression, find an equivalent expression with

the LCD. To do so, multiply by 1 using an expression for 1 made up

of factors of the LCD that are missing from the original

denominator.

c. Add the numerators. Write the sum over the LCD.

d. Simplify if possible.

 

LCMs of Algebraic Expressions

LCMs of Algebraic Expressions

 

To find the LCM of two or more algebraic expressions, we factor them. Then

we use each factor the greatest number of times that it occurs in any one expression.

 

Example

12x , 16y, 8xyz = ?

 

12x = 2. 2. 3 .x

16y = 2 . 2 . 2 . 2 .y

8xyz = 2 . 2 . 2 . x . y .z

 

LCM = 2 . 2 . 2 . 2 . 3 . x . y . z = 48 xyz

 

Adding Using the LCD

Adding Using the LCD

 

Let’s  finish adding \frac{5}{12} and \frac{7}{30}:

 

\frac{5}{12}+\frac{7}{30}=\frac{5}{2.2.3}+\frac{7}{2.3.5}.

 

The least common denominator, LCD, is 2 . 2 . 3 . 5. To get the LCD in the first denominator, we need a 5. To get the LCD in the second denominator, we need another 2. We get these numbers by multiplying by form of 1:

 

\frac{5}{12}+\frac{7}{30}=\frac{5}{2.2.3}.\frac{5}{5}+\frac{7}{2.3.5}.\frac{2}{2}   Multiplying by 1

= \frac{25}{2 . 2 .3.5}+\frac{14}{2.3.5.2} Each denominator is now the LCD.

=   \frac{39}{2.2.3.5}  Adding the numerators and keeping the LCD

= \frac{3.13}{2.2.3.5}  Factoring the numerator and removing a factor of 1: \frac{3}{3} = 1

= \frac{13}{20}.  Simplifying

 

Least Common Multiples

 Least Common Multiples

To add when denominators are different, we first find a common denominator. For a review, see Appendixes A and B. We know, for example, that to add \frac{5}{10} and \frac{7}{30}, we first look for the least common multiple, LCM. Of both 12 and 30. That number becomes the least common denominator, LCD. To find the LCM of 12 and 30, we factor.

12 = 2 . 2 . 3 ;

30 = 2 . 3 . 5.

The LCM is the number that has 2 as a factor twice, 3 as a factor once, and 5 as a factor once:

LCM = 2 . 2 . 3 . 5 = 60.

 

FINDING LCMS

To find the LCM, use each factor the greatest number of times that it appears in any one factorization.

 

 

Maximum–Minimum Problems

Maximum–Minimum Problems

 

We have seen that for any quadratic function f(x) = ax2 + bx + c, value of f(x) at the vertex is either a maximum or a minimum, meaning that either all outputs are smaller than that value for a maximum or larger than that value for a minimum.

 minimum and maximum

There are many types of applied problems in which we want to find a maximum or minimum value of a quadratic function can be used as a model, we can find such maximums or minimums by finding coordinates of the vertex.

 

Rational Expressions and Replacements

Rational Expressions and Replacements

 

Rational numbers are quotients of integers. Some examples are

 

\frac{2}{3} , \frac{4}{-5} , \frac{-8}{17}.

The following are called rational expressions or fraction expressions. They are quotients, or ratios, of polynomials:

 

\frac{3}{4}, \frac{z}{6}, \frac{3}{2 + x}

 

A rational expression is also a division. For example,

 

\frac{3}{4} means 3 ÷ 4 and \frac{x-8}{x+2} means (x-8) ÷ (x+2).

 

Because rational expressions indicate division, we must be careful to avoid denominators of zero. When a variable is replaced with a number that produces a denominator equal to zero, the rational expression is not defined.