Properties of integers - absolute value, comparing, substraction, addition, double negative

•  The absolute value of a number is the distance the number is away from 0 on a number line

•  The absolute value of x is written in symbols as |x|; |x| = x if x > 0 or x = 0 and |x| = -x if x < 0; that is, the absolute value of a number is always the positive value of that number

EX: |6| = 6 and |-6| = 6; the answer is positive 6 in both cases; this means the 6 and -6 are both 6 units away from 0 on a number line

•  A number line can be used to help compare and order integers; on a horizontal number line, integers to the right are greater than integers to the left; when comparing two integers, use the symbols < (less than) and > (greater than); the symbol points to the lesser number

•  If the signs of the numbers are the same, add; the answer has the same sign as the numbers
EX: (-4) + (-9) = -13 and 5 + 11 = 16

•  If the signs of the numbers are different, subtract; the answer has the sign of the number with the greater absolute value

EX: (-4) + (9) = 5 and (4) + (-9) = -5

•  Change subtraction to addition of the opposite number; a - b = a + (-b); that is, change the subtraction sign to addition and also change the sign of the number directly after the subtraction sign to the opposite; then follow the addition rules

EX: (8) - (12) = (8) + (-12) = -4 and (-8) - (12) = (-8) + (-12) = -20 and (-8) - (-12) = (-8) + (12) = 4; note that the sign of the number in front of the subtraction sign never changes

•  Multiply or divide, then follow these rules to determine the sign of the answer:

•  If the numbers have the same signs, the answer is positive

•  If the numbers have different signs, the answer is negative

•  It makes no difference which number is greater when you are trying to determine the sign of the answer

EX: (-2)(-5) = 10 and (-7)(3) = -21 and (-18) + (6) = -3

•  -(-a) = a; that is, the sign in front of the parentheses changes the sign of the contents of the parentheses

EX: -(-3) = 3 and -(3) = -3

•  -(a + b) = -a + (-b) and -(a - b) = -a - (-b) = -a + b; that is, when a negative sign is in front of a quantity, the negative sign is distributed to each term in the quantity

EX: -5(a + 6) = -5a - 30 and -2(x - 3) = -2x + 6

•  When a negative number is raised to a power, the negative number must be in parentheses; the values -ax and (-a)^x are not the same; in -a^x, only a is raised to the power of x, and the final value is negative; in (-a)^x, the final value is positive if x is even and negative if x is odd EX: -3⁴ = -81 and (-3)⁴ = 81