The absolute value of a number is its distance from zero on the number line, regardless of direction. It is always non-negative. The absolute value of a number x is written as |x|. For example, |7| = 7 and |–7| = 7, because both 7 and –7 are 7 units from zero.
Absolute value plays an important role in both mathematical theory and problem solving, especially in equations involving distance and size. It has four key properties that help define how it behaves:
- Non-negativity: |a| ≥ 0: The absolute value is never negative.
- Positive-definiteness: |a| = 0 if and only if a = 0: Zero is the only number with an absolute value of zero.
- Multiplicativeness: |ab| = |a||b|: The absolute value of a product equals the product of the absolute values.
- Subadditivity (Triangle Inequality): |a + b| ≤ |a| + |b|: The absolute value of a sum is less than or equal to the sum of the absolute values.
These properties are useful in algebra, geometry, and calculus, and they form the basis for many inequalities and distance-based reasoning.
How to Solve Absolute Value Equations
To solve absolute value equations, follow these steps:
- Isolate the absolute value expression on one side of the equation.
- Set up two cases: one where the expression equals the positive value, and one where it equals the negative.
- Solve each case separately.
- Check your solutions in the original equation—especially when variables appear outside the absolute value, since some solutions might not work.
Let’s work through solving an example absolute value equation!
Absolute Value Equations
An absolute value equation contains an expression within absolute value symbols. For example:
|x – 3| = 5
This asks, “What values of x are 5 units away from 3?” To solve, you consider both the positive and negative cases:
- x – 3 = 5 → x = 8
- x – 3 = –5 → x = –2
So the solutions are x = –2 or x = 8.
Understanding absolute value and how to solve absolute value equations is essential for working with distances, constraints, and piecewise-defined functions in algebra and beyond.
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