Absolute Value

Algebra

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) $| - 5 | = 5$

B. Equation related absolute value

$| x | = a for a \geq 0$

Method 1

$| x | = a$

$x = - a atau x = + a$

Method 2

$| x | = a$

$x^{2} = a^{2}$

$x^{2} - a^{2} = 0$

$(x + a) (x - a) = 0$

$x = - a atau x = + a$

Example 02

Find the value of $x$ where $| x | = 3$

Method 1

$| x | = 3$

$x = - 3 atau x = 3$

Method 2

$| x | = 3$

$x^{2} = 3^{2}$

$x^{2} - 3^{2} = 0$

$(x + 3) (x - 3) = 0$

$x = - 3 atau x = 3$

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 $\| x \| > 5$ $x^{2} > 5^{2}$ $x^{2} - 5^{2} > 0$ $(x + 5) (x - 5) > 0$

| 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

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