Algebra With Integers

The set of integers, Z, consists of the whole numbers and their negative counterparts. Z = { …, -3, -2, -1, 0, 1, 2, 3, … }
The absolute value of a number is the distance between the number and zero on a number

It is defined by the formula: x  =        x,  if x ≥ 0
− x,  if x < 0

The algebra with integers is the set of integers that has whole number and their negative counterparts. The algebra with integers include different operations as addition, subtraction, multiplication, division of algebra with integers.Let us see the algebra with integers concepts and example problems.

Algebra with integers:

Algebra with Integers Operation Rules:

Add:

To add two integers which have the same sign, add their absolute values and give the result the same sign as the integers.
To add two integers with opposite signs, subtract the smaller absolute value from the larger and give the result the sign of the integer with the larger absolute value. If the signs are same then add the numbers and keep the sign same as signs of  numbers given.

Example:  - 5 + (-6) = - 11,   7 + 8 = 15

If the signs are different, subtract the numbers and use the sign of the larger number.

Example:   -9 + 4 = -5,       -3 + 12 = 9

Subtract:

To subtract two integers change the subtraction symbol to addition while reversing the sign of the second integer. Here  use the rules of addition. Change the signs, or add the opposite, then follow the addition rules.

Example:  -7 – 10 = -7 + (-10) = -17,      -5 – (-8) = -5 + 8 = 3

Multiply/Divide

If the signs are the same then the answer is positive.

Multiplying Integers

To multiply two integers, first multiply their absolute values. If the integers have the same sign then the result is positive, otherwise the result is negative.

Dividing Integers

To divide two integers, first divide their absolute values. If the integers have the same sign then the result is positive, otherwise the result is negative.

Example: 3 x 6 = 18,  -3 x (-6) = 18,  `18/6` = 3,   `(-18)/(-6)` = 3

Algebra with integers:

How to solve one step equations:

Rewrite the equation to add, subtract, multiply or divide the opposite number to each side.

Examples:

2X = 7        2X / 2 = 7 / 2            X/2 = 8

`X/2 * 2 = 8 * 2`

X – 7 = 10             X – 7 + 7 = 10 + 7

Evaluate

To evaluate, you replace the variable with a number In the expression, x + 8, if you put 2 in for x, the expression would be: 2 + 8.

How to add integers:

Look at the sign for each number.


Example: 6+ 6 = (The signs would be -)

If the signs are the same, add the two numbers and take that sign.


Example: 6 + 6 = 12,  6 + 6 = 12


If the signs are different, take the larger number minus the smaller number. The sign of the result would be sign of the larger number in the problem.


Example: -6 + 3 = 3 (6-3 =3 and 6 is the higher number so the answer is negative.)

How to subtract integers:

Look at the problem and find all of the signs that are side by side without a number in between the two signs.


Example: 3 + 6,   3-6 = 3 + 6,

If there is signs of two negatives side by side, change them into a positive. If sign of integer is a negative and a positive side by side, change them into a negative.


Example: -3 + -6,

Look at the sign for each number.


Example:  -6-6 = (The signs would be –  )

If the signs are the same, add the two numbers and take that sign.


Example: 6 -6 = 12

If the signs are different, minus the larger number from the smaller number. The sign for the answer is the sign of the higher number of the given  problem.


Example: 6 - 3 = 3 (6-3=3 and 6 is the higher number so the answer is positive.)

How to multiply and divide integers:

Look at the signs of each number.


If the signs are the same, multiply or divide the numbers and your answer would be positive.


Examples: 6 * 3 = 18, 5 * 3 = 15, 24 / 6 = 4, 24 / 6 = 4

If the signs are different, multiply or divide the numbers and your answer would be negative.


Examples: 6 * 3 = 18, 5* 3 = 15, 24/ 6 = 4, 24 / 6 = 4

A symbol that stands for a number

In the algebraic problem x + 6, x is called a variable because it can stand for any number.


The operation used to make zero or one

Example: In the equation X + 6, the inverse operation would be to subtract 6, since you first added six.