# Algebra

### 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

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