All real numbers correspond to points on the number line and all points on the number line correspond to real numbers. All real numbers except zero are either positive or negative.
On a number line, numbers corresponding to points to the left of zero are negative and numbers corresponding to points to the right of zero are positive. For any two numbers on the number line, the number to the left is less than the number to the right; for example,
-4 < -3 < -3/2 < -1 and 1 < √2 < 2
Absolute value of any real number is the same number without a negative sign. It is the distance between a number and zero on the number line. The symbol for absolute value is | |. Absolute value of any non-zero number is positive.
Thus, 3 and -3 have the same absolute value, 3, since they are both three units from zero. The absolute value of 3 is denoted |3|. Examples of absolute values of numbers are:
|-5| = |5| = 5 and |0| = 0
Remember that absolute value pertains only to the value contained within the absolute value bars. So if you see a negative sign outside the bars, the resulting value is negative. For example, – | –3 | = –3 because although the absolute value of –3 is 3, the negative sign outside the bars makes the end result a negative.
When you are working with variables in absolute-value expressions, remember that there is likely more than one solution for the variable because the value within the absolute value sign may be positive or negative.
If x, y, and z are real numbers, then
x + y = y + x and xy = yx
(x + y) + z = x + (y + z) and (xy)z = x(yz)
x(y + z) = xy + xz
If x and y are both positive, then x + y and xy are positive
If x and y are both negative, then x + y is negative and xy is positive
If x is positive and y is negative, then xy is negative
If xy = 0, then x = 0 or y = 0
| x + y | ≤ | x | + | y |