Absolute Value
IMAT Interactive Study Tool
What is Absolute Value?
The absolute value of a number is its distance from zero on the number line. Since distance cannot be negative, the absolute value of a number is always non-negative (either positive or zero).
It is denoted by two vertical bars around the number, like \(|x|\).
As you can see, both 3 and -3 are a distance of 3 from zero. So, \(|3| = 3\) and \(|-3| = 3\).
Formal Definition
Mathematically, the absolute value is defined as a piecewise function:
This definition just means:
- If the number inside is already positive or zero, do nothing.
- If the number inside is negative, make it positive by multiplying by -1. (e.g., \(|-5| = -(-5) = 5\)).
1. True or False: The absolute value of any non-zero number is always positive.
2. True or False: \(|x|\) is always equal to \(x\).
Example 1: Solving an Absolute Value Equation
Solve for \(x\) in the equation \(|2x - 1| = 7\).
Step 1: Understand the core concept.
The expression inside the absolute value bars, \(2x - 1\), must have a distance of 7 from zero. This means the expression itself can be either 7 or -7.
Step 2: Set up two separate linear equations.
Case 1: \(2x - 1 = 7\)
Case 2: \(2x - 1 = -7\)
Step 3: Solve each equation for \(x\).
Case 1: \(2x = 8 \implies x = 4\)
Case 2: \(2x = -6 \implies x = -3\)
Conclusion: The two possible solutions are \(x = 4\) and \(x = -3\).
Example 2: Evaluating an Expression
Evaluate the expression \(|5 - 12| - |-3|\).
Step 1: Evaluate the inside of each absolute value separately.
First term: \(|5 - 12| = |-7|\)
Second term: \(|-3|\)
Step 2: Apply the definition of absolute value to each term.
First term: \(|-7| = 7\)
Second term: \(|-3| = 3\)
Step 3: Perform the final subtraction.
\(7 - 3 = 4\)
Conclusion: The value of the expression is 4.
| Property | Formula | Explanation |
|---|---|---|
| Non-negativity | \(|a| \ge 0\) | Absolute value is always positive or zero. |
| Symmetry | \(|a| = |-a|\) | A number and its opposite have the same distance from zero. |
| Multiplication | \(|ab| = |a||b|\) | The absolute value of a product is the product of the absolute values. |
| Triangle Inequality | \(|a+b| \le |a|+|b|\) | The absolute value of a sum is less than or equal to the sum of the absolute values. |