Introduction to Absolute Value

The absolute value of a number tells you how far that number is from zero on the number line, no matter which direction you travel to get there. Because distance is never negative, the absolute value of any real number is always zero or positive. This one idea, "how far from zero," is the whole concept behind absolute value.

Absolute value is written using two vertical bars around a number or expression, called the absolute sign. For example, \(|7|\) is read as "the absolute value of 7," and \(|-7|\) is read as "the absolute value of negative 7." Even though 7 and negative 7 sit on opposite sides of zero, they are the same distance away, so:

\(|7| = 7\) and \(|-7| = 7\)

More formally, the absolute value of a number \(x\), written \(|x|\), follows this rule:

\(|x| = x\) when \(x \ge 0\)

\(|x| = -x\) when \(x < 0\)

This looks more complicated than it is. All it says is: if the number is already zero or positive, leave it alone. If the number is negative, drop the negative sign to get its positive counterpart. Either way, the answer describes a distance, so it can never come out negative.

Example 1: Find the absolute value of 15.

15 is already positive, so \(|15| = 15\).

Example 2: Find the absolute value of negative 100.

\(-100\) is negative, so drop the negative sign: \(|-100| = 100\).

Example 3: Evaluate \(|4 - 9|\).

Work out what's inside the bars first: \(4 - 9 = -5\). Then take the absolute value: \(|-5| = 5\). So \(|4 - 9| = 5\), not \(-5\).

Example 4: Find the absolute value of each integer: \(-6\), \(0\), and \(8\).

\(|-6| = 6\), \(|0| = 0\), and \(|8| = 8\). Notice that zero is the only number whose absolute value equals itself and stays at zero, since it is exactly zero units from zero.

Absolute value can also be viewed as a function, \(f(x) = |x|\), which produces the distinctive V-shaped graph below. Every output value is zero or positive, matching the rule you just learned, and the graph is symmetric about the y-axis because \(|x|\) and \(|-x|\) always give the same result.

This shape and its properties are explored in more depth in the lesson on absolute value functions.

Some students assume the absolute sign simply "makes everything positive" without thinking about what's happening inside the bars first. Always simplify the expression inside the bars before applying the absolute value, as in Example 3 above. Also remember that \(-|x|\) (a negative sign placed outside the bars) is not the same as \(|-x|\); the outside negative sign is applied after the absolute value is taken, so \(-|5| = -5\).

Once you're comfortable finding the absolute value of a number, the next step is usually applying it inside algebraic statements. This idea is central to solving absolute value equations, where you must consider both the positive and negative case, and to solving absolute value inequalities, where the same distance idea defines a range of solutions rather than a single number.

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