What is absolute value?
Put a number on the number line. Count the steps to 0. That count is its absolute value. We write it with two bars: |x|.
So |5| = 5 and |-5| = 5. Both are 5 steps from 0. |0| = 0.
In symbols: |x| = x if x is 0 or positive, and |x| = -x if x is negative. The "minus" in -x turns a negative number into a positive one, so |x| is never below 0.
Distance between two numbers: |x - a|
The distance between x and a on the line is |x - a|. It is the same as |a - x|. Example: the distance from 2 to 9 is |9 - 2| = 7. From -4 to 3 it is |3 - (-4)| = 7.
Remember: |x| is just |x - 0|, the distance from 0.
Solving |x + a| = b
Read |x - c| = b as: "x is b steps from c". Go b steps right and b steps left from c. There are two answers: x = c + b and x = c - b.
For |x + 3| = 5: write it as |x - (-3)| = 5. So x = -3 + 5 = 2 or x = -3 - 5 = -8. Check: |2 + 3| = 5 and |-8 + 3| = 5.
If b is negative, there is no answer, because a distance cannot be negative. If b = 0, there is one answer: x = c.
Intervals on the number line
An interval is all the numbers between two ends.
- [a, b] closed: a and b are included (filled dots). It means a ⤠x ⤠b.
- (a, b) open: a and b are not included (empty dots). It means a < x < b.
- [a, b) and (a, b] are half open.
- (a, â): everything above a. The sign â means "goes on for ever". It always gets a round bracket.
Link to absolute value: |x - c| < r means c - r < x < c + r, the interval (c - r, c + r). And |x - c| > r means x is outside it: x < c - r or x > c + r.
Try it: walk the line
Draw a line on the floor with chalk and mark -5 to 5. Stand on 0. Take 4 steps right, then go back to 0 and take 4 steps left. Where are you? 4 and -4. Both are |x| = 4. Now guess before you slide: where must x be so that |x - 2| = 3? Check in the 3D.
Key formulas and definitions
- |x| = x if x âĨ 0; |x| = -x if x < 0
- |x - a| = distance between x and a
- |x - c| = b (b > 0) â x = c + b or x = c - b
- |x - c| < r â c - r < x < c + r
- |x - c| > r â x < c - r or x > c + r
- [a, b] includes both ends; (a, b) includes neither
Worked examples
1. Find |-7|, |4| and |0|.
|-7| = 7 (7 steps from 0). |4| = 4. |0| = 0.
2. Find the distance between -2 and 5.
|5 - (-2)| = |7| = 7.
3. Solve |x| = 6.
x is 6 steps from 0, so x = 6 or x = -6.
4. Solve |x + 4| = 3.
Write as |x - (-4)| = 3. x = -4 + 3 = -1 or x = -4 - 3 = -7. Check: |-1 + 4| = 3 and |-7 + 4| = 3.
5. Write 2 < x ⤠9 as an interval and say which dots are filled.
(2, 9]. The dot at 2 is empty (2 is not included). The dot at 9 is filled.
6. Solve |2x - 5| = 7.
2x - 5 = 7 gives x = 6. 2x - 5 = -7 gives x = -1. Check: |12 - 5| = 7 and |-2 - 5| = 7.
7. Write |x - 1| < 3 as an interval.
1 - 3 < x < 1 + 3, so (-2, 4).
8. Solve |x - 2| = |x + 6|.
x is the same distance from 2 and from -6, so it is the middle: (2 + (-6)) / 2 = -2. Check: |-4| = 4 and |4| = 4.
Common mistakes
- Writing |-5| = -5. Absolute value is never negative: |-5| = 5.
- Giving only one answer for |x - c| = b. There are two places, c + b and c - b.
- Putting a square bracket next to â. Infinity is not a number, so always use â).
- Solving |x + 3| = 5 as x = 3 + 5. The centre is -3, not 3. Write x + 3 as x - (-3).