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# absolute value

The absolute value of a real number is its distance from zero on the number line, regardless of direction. It is represented by two vertical bars on either side of the number (e.g., |x|). The absolute value of a number is always non-negative, as it reflects magnitude without regard to direction. For any real number xx, the absolute value is defined as:

• âˆ£xâˆ£=xâˆ£xâˆ£=x if xâ‰¥0xâ‰¥0
• âˆ£xâˆ£=âˆ’xâˆ£xâˆ£=âˆ’x if x<0x<0

Key Concepts:

• Non-Negativity: The absolute value of a number is always positive or zero because it represents distance, which cannot be negative.
• Zeroâ€™s Absolute Value: The absolute value of zero is zero (|0| = 0), as zero is neither positive nor negative and represents no distance from itself.
• Symmetry: The absolute value function is symmetric about the y-axis on a graph, indicating that |x| = |-x| for any real number xx.

Examples:

1. Positive Number: The absolute value of 5 is 5.âˆ£5âˆ£=5âˆ£5âˆ£=5Explanation: Since 5 is already positive, its absolute value is itself.
2. Negative Number: The absolute value of -3 is 3.âˆ£âˆ’3âˆ£=3âˆ£âˆ’3âˆ£=3Explanation: The distance of -3 from 0 is 3 units, hence its absolute value is 3.
3. Zero: The absolute value of 0 is 0.âˆ£0âˆ£=0âˆ£0âˆ£=0Explanation: Zero is at no distance from itself, so its absolute value is 0.
4. Variable Expression: For a negative variable, such as x=âˆ’8x=âˆ’8, the absolute value of xx is 8.âˆ£âˆ’8âˆ£=8âˆ£âˆ’8âˆ£=8Explanation: The distance of -8 from 0 on the number line is 8 units.
5. Use in Equations: Solving âˆ£xâˆ’4âˆ£=2âˆ£xâˆ’4âˆ£=2 results in two possible equations: xâˆ’4=2xâˆ’4=2 and xâˆ’4=âˆ’2xâˆ’4=âˆ’2, yielding x=6x=6 or x=2x=2.