The absolute value of a rational number x, written as |x|, represents its distance from 0 on the number line. |53| = 53, |−53| is also equal to 53 and |0| = 0.

Step-by-step solution

Idea: Absolute value = distance from 0. A distance is never negative, and a number and its negative sit at the same distance on opposite sides of 0.

1. 53 = 123 lies 53 units to the right of 0, so |53| = 53.½ mark
2. −53 lies 53 units to the left of 0. The direction does not matter for a distance, so |−53| = 53.½ mark
3. 0 is at distance 0 from itself, so |0| = 0.½ mark
4. So the absolute value of a positive number is the number itself, of a negative number is its positive value, and |x| ≥ 0 always.½ mark

Check: Distance between 5/3 and −5/3 is 53 + 53 = 103 = |53 − (−53)| ✓.

Answer to write in the exam

Common mistakes that cost marks

• Writing |−53| = −53. An absolute value is a distance and is never negative.
• Thinking |x| just ‘removes the minus sign’ from the symbol; |−(−2)| = |2| = 2, so simplify inside first.
• Writing |0| = 1 or saying |0| is undefined.

How this can come in the exam

Try one yourself

Find |−94|, |27| and |−6|.

More questions like this

• Try to explain why the average of two rational numbers a and b, which equals (a + b)2, is always a rational number between a and b.
• This means that there are infinitely many rational numbers between any two points. It feels as though the rational numbers must completely fill the number line, leaving no gaps whatsoever. But do they?
• Represent the rational numbers 23, −54 and 112 on a single number line.
• Find three distinct rational numbers that lie strictly between −12 and 14.
• Simplify the expression: (−14) + (512).

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