The absolute value of an expression represents the positive distance from the expression to the number 0. Absolute value can be thought of as a positive output operator as its result is always positive:

Consequently, problems involving absolute values must be set up differently than those without absolute value:

is different from because both positive and negative 4 equal 4 when inside of the absolute value operator:

So, the original equation, , involving the absolute value needs to be set up as 2 equations:

and

Plugging these values back in to verify:

and

The number line contains 0 and extends outward to the left to negative infinity and to the right to positive infinity. As one moves from left to right on the number line, the value of the numbers increase:

As one moves from right to left on the number line, the numbers decrease:

When ordering positive and negative integers from smallest to largest, begin with the negative number of the largest magnitude ( has a larger magnitude than ) and list the remaining negative integers in decreasing magnitude until 0 is reached. Continue ordering the positive integers from smallest to largest.

Rational numbers are those that can be expressed as a fraction of 2 integers. These numbers can be directly compared by dividing the numerator by the denominator and converting the fraction to a decimal:

Rational numbers containing a negative sign in the numerator or the denominator can be rewritten with the negative sign in front of the fraction: