Page 3 - (1-1)eng

Elly fat mn 3omry ( 1 – 1 )
The Absolute Value

The absolute value of a real number x is the distance between that number x and 0 on the number

line. The absolute value of a real number x is written | x |.
| – 5 | = 5 , which is the distance between – 5 and 0

5 units

– 7 – 6 – 5 – 4 – 3 – 2 – 1 0 1 2 3 4 5 6 7 8
4 units
| 4 | = 4 , which is the distance between 4 and 0
  x x  0
In general | x | = 
 x x  0
– x is not a negative number in this case because x is a negative number, and the
negative of a negative number is positive;

for example,
If x = – 2, then – x = – ( – 2 ) = 2

Notes  | x |  0
 | – x | = | x |
2
2
2
 | x | = | x | = x
 x 3 x  0
3
3
| x | = | x | = 
 x 3 x  0
2
 x = | x |

x x
 | x y | = | x | | y | , = , y  0
y y
Absolute – Value Equations
 | x | = 0  x = 0
 | x | = a , a > 0  x = a or x = – a
 | x | = | y |  x = y or x = – y

Absolute – Value Inequalities
 x  = 2  x  = 2

– 3 – 2 – 1 0 1 2 3
 x  > 2  x  < 2  x  > 2
If a > 0 , then
  x  < a  – a < x < a
 x   a  – a  x  a
  x  > a  x < – a or x > a
 x   a  x  – a or x  a

Calculus 2 Unit (1)