MATH

TYPES OF NUMBERS

INtEGERs are whole numbers, including the counting numbers, the negative counting numbers, and zero. 3, 2, 1, 0, -1, -2, -3 are examples of integers. RATIONAL NUMBERs are made by dividing one integer by another integer. They can be expressed as fractions or as decimals. 3 divided by 4 makes the rational number ¾ or 0.75. IRRATIONAL NUMBERS are numbers that cannot be written as fractions; they are decimals that go on forever without repeating.The number r (3.14159...) is an example of an irrational number. IMAGINARY NUMBERS are numbers that, when squared, give a negative result. Imaginary numbers use the symbol i to represent

V (-1), so 31 = 3V (-1) and (3) 2 = -9. COMPLEX NUMBERS are

combinations of real and imaginary numbers, written in the form a + bi, where a is the real number and b is the imaginary number.

An example of a complex number is 4 + 2i. When adding complex numbers, add the real and imaginary numbers separately: (4 + 2i) + (3 + i) = 7 + 3i.

Examples

Is v5 a rational or irrational number?

V5 is an irrational number because it cannot be written as a fraction of two integers. It's a decimal that goes on forever without repeating.

What kind of number is —v64?

-V64 can be rewritten as the negative whole number -8, so it's an integer.

3. Solve (3 + 5i) - (1 - 2i)

Subtract the real and imaginary numbers separately.

3-1 = 2

5i - (-2i) = 5i + 2i = 7i

So (3 + 5i) - (1 - 2i) = 2 + 7i

POSITIVE AND NEGATIVE NUMBER RULES

Adding, multiplying, and dividing numbers can yield positive or negative values depending on the signs of the original numbers.

Knowing these rules can help determine if your answer is correct.

(+) + (-) = the sign of the larger number

(-) + (-) = negative number

(-) x (-) = positive number

(-) × (+) = negative number (-) ÷ (-) = positive number

(-) + (t) = negative number

Examples

1. Find the product of -10 and 47.

（一）（十）（一）

-10 × 47 = -470

2. What is the sum of -65 and -32?

(-) + (-) = (-)

-65 + -32 = -97

3. Is the product of -7 and 4 less than -7, between -7 and 4, or greater than 4?

（）（十）（一）

-7 x 4 = -28, which is less than -7

4. What is the value of -16 divided by 2.5?

(-) ÷ (+) = (-)

-16 ÷ 2.5 = -6.4

ORDER OF OPERATIONS

Operations in a mathematical expression are always performed in a specific order, which is described by the acronym PEMDAS:

Parentheses
Exponents
Multiplication
Division
Addition
Subtraction

Perform the operations within parentheses first, and then address any exponents. After those steps, perform all multiplication and division. These are carried out from left to right as they appear in the problem.

Finally, do all required addition and subtraction, also from left to right as each operation appears in the problem.

Examples

1. Solve: [-(2)2 - (4 + 7)]

First, complete operations within parentheses:

-(2)2 - (11)

Second, calculate the value of exponential numbers:

-(4) - (11)

Finally, do addition and subtraction:

-4 - 11 = -15

2. Solve: (5)2 ÷ 5 + 4x2
First, calculate the value of exponential numbers:
(25) ÷ 5+4×2
Second, calculate division and multiplication from left to right:
5+8
Finally, do addition and subtraction:
5 + 8 = 13

3. Solve the expression: 15 x (4 + 8) - 33
First, complete operations within parentheses:
15 × (12) - 33
Second, calculate the value of exponential numbers:
15 × (12) - 27
Third, calculate division and multiplication from left to right:
180 - 27
Finally, do addition and subtraction from left to right:
180 - 27 = 153

GREATEST COMMON FACTOR

The greatest common factor (GCF) of a set of numbers is the largest number that can evenly divide into all the numbers in the set. To find the GCF of a set, find all the factors of each number in the set.

A factor is a whole number that can be multiplied by another whole number to result in the original number. For example, the number 10 has four factors: 1, 2, 5, and 10. (When listing the factors of a number, remember to include 1 and the number itself.) The largest number that is a factor for each number in the set is the GCF.

Examples

1. Find the greatest common factor of 24 and 18.

Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24

Factors of 18: 1, 2, 3, 6, 9, 18

The greatest common factor is 6.

2. Find the greatest common factor of 121 and 44.

Since these numbers are larger, it's easier to start with the smaller number when listing factors.

Factors of 44: 1, 2, 4, 11, 22, 44

Now it's not necessary to list all the factors of 121. Instead, we can eliminate those factors of 44 that do not divide evenly into 121:

121 is not evenly divisible by 2, 4, 22, or 44 because it's an odd number. This leaves only 1 and 11 as common factors, so the GCF is 11.

3. First aid kits are being assembled at a summer camp. A complete first aid kit requires bandages, sutures, and sterilizing swabs, and each of the kits must be identical to other kits. If the camp's total supplies include 52 bandages, 13 sutures, and 39 sterilizing swabs, how many complete first aid kits can be assembled without having any leftover materials?

This problem is asking for the greatest common factor of 52, 13, and 39. The first step is to find all the factors of the smallest number, 13.

Factors of 13: 1, 13

13 is a prime number, meaning that its only factors are 1 and itself. Next we check to see if 13 is also a factor of 39 and 52:

13 × 2 = 26

13 × 3 = 39

13×4 = 52

We can see that 39 and 52 are both multiples of 13. This means that 13 first aid kits can be made without having any leftover materials.

ALGEBRA

Algebraic expressions and equations include a VARIABLE, which is

a letter standing for a number. These expressions and equations

are made up of TERMS, which are groups of numbers and variables

(e.g., 2xy). An EXPRESSION is simply a set of terms (e.g., 3x + 2xy),

while an EQUATION includes an equal sign (e.g., 3x + 2xy = 17).

When simplifying expressions or solving algebraic equations, you'll

need to use many different mathematical properties and operations,

including addition, subtraction, multiplication, division, exponents,

roots, distribution, and the order of operations.

ALGEBRAIC EXPRESSIONS

Evaluating Algebraic Expressions

To evaluate an algebraic expression, simply plug the given value(s)

in for the appropriate variable(s) in the expression.

Example

Evaluate 2x + 6y - 3z, if x = 2, y = 4, and z=-3.

Plug in each number for the correct variable and simplify:

2x + 6y - 3z= 2(2) + 6(4) - 3(-3) = 4 + 24 + 9 = 37

Adding and Subtracting Terms

Only LIKE TERMS, which have the exact same variable(s), can be

added or subtracted. CONSTANTS are numbers without variables

attached, and those can be added and subtracted together as well.

When you simplify an expression, like terms should be added or

subtracted so that no individual group of variables occurs in more

than one term. For example, the expression 5x + 6xy is in its simplest

form, while 5x + 6xy - 11xy is not because the term xy appears

more than once.

multiply, or divide whatever you want as long as you do the same

thing to both sides.

same basic steps:

Most equations you'll see on the GED can be solved using the

1. Distribute to get rid of parentheses.

2. Use the least common denominator to get rid of

fractions.

3. Add/subtract like terms on either side.

4. Add/subtract so that constants appear on only one side

of the equation.

5. Multiply/divide to isolate the variable.

Examples

1. Solve for x: 25x + 12 = 62.

This equation has no parentheses, fractions, or like terms

on the same side, so you can start by subtracting 12 from

both sides of the equation:

25x + 12 = 62

(25x + 12 - 12 = 62 - 12

25x= 50

Now divide by 25 to isolate the variable:

x=2

2. Solve the following equation for x: 2x - 4(2x + 3) = 24.

Start by distributing to get rid of the parentheses (don't

forget to distribute the negative):

2x - 4(2y + 3) = 245

2x - 8x - 12 = 74

There are no fractions, so now you can join like terms:

2x - 8x - 12 = 24 >

-6x-12 = 74

Now add 12 to both sides and divide by -6.

-6x -12 = 24

(-6y - 12) + 12 = 24 + 12 >

-6x = 36 -

Or - 30

x=-6

3. Solve the following equation for x: x/3 + 1/2 = x/6 - 5/12

Start by multiplying by the least common denominator to

get rid of the fractions: x/3 + 1/2 = x/6 - 5/12

Inner: 2 × 3x = 6x

Last: 2 x 3 = 6

Now combine like terms:

15x2 + 21x + 6

4. Simplify the expression:

2xty°z

Simplify by looking at each variable and crossing out those

that appear in the numerator and denominator:

Factoring Expressions

FACTORING is splitting one expression into the multiplication of

two (or more) expressions. It requires finding the HiGHEST COMMON

PACTOR and dividing terms by that number. For example, in the

expression 15x + 10, the highest common factor is 5 because boch

ems are divisible by 5: IS* = 3x and I0 = 2. When you factor the

expression, you get 5(3x + 2).

Sometimes, it's difficult to find the highest common factor.

In these cases, consider whether the expression fits a polynomial

identity. A polynomial is an expression with more than one term. If

you can recognize the common polynomials listed below, you can

easily factor the expression.

7 - b = (a + b)(a - b)

a + 2ab + b3 = (a + b) (a + b) = (a + b)2

a - 2ab + b = (a - b) (a - b) = (a - b)2

a + b' = (a + b) (a - ab - b2)

à - b° = (a - b) (a + ab + b2)

Examples

1. Factor the expression 27x - 9x.

First, find the highest common factor. Both terms are

divisible by 9:

1 = 3x and * =x

Now the expression is 9(3x2 -x)-but wait, you're not

done! Both terms can be divided by x:

3* =3x and {= 1.

The final factored expression is 9x(3x - 1).

Example

simplify the expression 5xy +/y + 2yz + 11xy - 5yz.

Start by grouping together like terms;

(5xy + 11xy) + (2yz - 5yz) + 7y

Now you can add together each set of like terms:

16xy + 7y - 3yz

Multiplying and Dividing Terms

To multiply a single term by another, simply multiply the coe:

Talents and then multiply the variables. Remember that when

you multiply variables with exponents, those exponents are added

together. For example, (y) (Ny) = sy.

When multiplying a term by a set of terms inside parentheses,

you need to DISTRIBUTE to each term inside the parentheses as

shown below:

a(b+c) = ab + ac

Figure 2.1. Distribution

When variables occur in both the numerator and denominator

of a fraction, they cancel each other out. So a fraction with variables

in its simplest form will not have the same variable on the top and

Bottom.

Examples

1. Simplify the expression (3xty?z)(2y'z°).

Multiply the coefficients and variables together:

3x2=6

xy=y

Now put all the terms back together:

6x*y°z°

2. Simplify the expression: (2y2) (y] + 2xy?z + 4z).

Multiply each term inside the parentheses by the term Zy:

(2y*) (v° + 2xy-z + Az)

(2y? x y*) + (2y° × 2xyz) × (2y? × 42)

2ys + 4xy'z + 8y?z

3. Simplify the expression: (5x + 2) (3x + 3).

Use the acronym FOIL--First, Outer, Inner, Last--to

multiply the terms:

First: 5x × 3x = 15x2

Outer: 5x × 3 = 15x

SYSTEMS OF EQUATIONS

A system of equations is a group of related question each of which has the same variable. The problems you can the GED will most

has the variable two equations has two variables, although

Joe Valh-sio solve sets of counting number of variable

X long as there are a corresponding number of equations (Solve x system with four variables, you need four equations),

"There are two main methods used to solve systems of equation,

In subsTituTioN, one equation is solved for a single variable, and

lIne veiling expression is substituted into the second equation for

the correct variable. In HIMINATION, equations are added together

so that variables cancel and one variable can be solved for.

Examples

1. Solve the following system of equations: 3y - 4+*=0

and 5x + 6y = 11.

To solve this system using substitution, first solve one

equation for a single variable:

3y -4+x=0

3y+x=4

x=4-3y

Next substitute the expression to the right of the equal

sign for x in the second equation:

5x + 6v= 11

5(4 - 3y) + 6v=11

20 - 15y + 6y = 11

20 - 9y= 11

-9y=-9

y=1

Finally, plug the value for y back into the first equation to

find the value of x:

3y -4+ x=0

3(1) -4+x=0

-1+x=0

x=1

The solution is x = 1 and y = 1, or the point (1, 1).

2. Solve the system: 2x + 4y = 8 and 4x + 2y = 10.

To solve this system using elimination, start by

manipulating one equation so that a variable (in this case

x) will cancel when the equations are added together:

2x + 4y = 8

-2(2x + 4y = 8)

-4x - 8y = -16

1= 745

Since the determinant is positive, we expect two real

numbers for x. Solve for the two possible answers:

x= 7+5 ) x=2

x=2-5>x=3

Graphing Quadratic Equations

Graphing a quadratic equation forms a PARABOLA. A parabola is a

Luteshoe shaped curve that is symmetrical about a vertical aris

passing through the veRTex (the highest or lowest point on the

parabola). Each term in the equation as + bx + c = 0 affects the

shape of the parabola. A bigger value for a makes the curve

while a smaller value makes the curve wider. A negative value for a

flips the parabola upside down. The axis of symmetry is the vertical

line x= Zi

-b. To find the y coordinate for the vertex, plug this value

for y into the expression ax? + bx + C.

axis of symmetry

vertex

Figure 2.1. Parabola

The easiest way to graph a quadratic equation is to find the ais

of symmetry, solve for the vertex, then create a table of points by

Pligsing in other numbers for sand solving for y. Plot these points

and trace the parabola.

FRACTION DECIMAL