Absolute Value
A. Absolute value
Absolute value of a number represented by \(| \: x \: | = a\), where \(a\):
-
- cannot be negative
- only has positive value or 0
Example 01
(A) \(| 0 | = 0\)
(B) \(| 5 | = 5\)
(C) \(|{\color {blue} -5 }| = 5\)
B. Equation related absolute value
\(|x| = a \text{ for } a \geq 0\)
Method 1
\( |x| = a \)
\( x = -a \text{ atau } x = +a \)
Method 2
\( |x| = a \)
\( x^2 = a^2 \)
\( x^2 - a^2 = 0 \)
\( (x + a)(x - a) = 0 \)
\(x = -a \text{ atau } x = +a \)
Example 02
Find the value of \(x\) where \(| x | = 3\)
Method 1
\( | x | = 3 \)
\( x = -3 \text{ atau } x = 3 \)
Method 2
\( | x | = 3 \)
\( x^2 = 3^2 \)
\( x^2 - 3^2 = 0 \)
\( (x + 3)(x - 3) = 0 \)
\( x = -3 \text{ atau } x = 3 \)
C. Inequalities related absolute value
| \(|x| < a \), for \(a > 0\) | |
|---|---|
| Method 1
\( -a < x < a \) Example \( |x| < 5 \) \( -5 < x < 5 \) |
Method 2
\( x^2 < a^2 \) Example \( |x| < 5 \) \( x^2 < 5^2 \) \( x^2 - 5^2 < 0 \) \( (x + 5)(x - 5) < 0 \) |
| \(|x| > a \), for \(a > 0\) | |
| Method 1
\( x < -a \) atau \( x > a \) Example \( |x| > 5 \) \( x < -5 \) or \(x > 5 \) |
Method 2
\( x^2 > a^2 \) Example \( |x| > 5 \) \( x^2 > 5^2 \) \( x^2 - 5^2 > 0 \) \( (x + 5)(x - 5) > 0 \) |
Exercise