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ABSOLUTE VALUE
In this worksheet you will get known a new term. It is an absolute value. You will also try to give
a definition of this term. In next worksheet you will learn, how to work with an absolute value function.
Geometrical interpretation of absolute value
Before making a definition of absolute value we will first try to explain meaning of absolute value on
a numerical axis.
1. Mark following numbers on the numerical axis.
1; -1; 4; -4; 7; -7; 5,5;-5,5
2. Comlete these statements.
a) The distance of number 1 from the coordinate origin is/are …… unit/units.
b) The distance of number -1 from the coordinate origin is/are …… unit/units.
c) The distance of number 4 from the coordinate origin is/are …… unit/units.
d) The distance of number -4 from the coordinate origin is/are …… unit/units.
e) The distance of number 7 from the coordinate origin is/are …… unit/units.
f) The distance of number -7 from the coordinate origin is/are …… unit/units.
g) The distance of number 5,5 from the coordinate origin is/are …… unit/units.
h) The distance of number-5,5 from the coordinate origin is/are …… unit/units.
Absolute value of real number is conected with the distance of this real number from zero.
ABSOLUTE VALUE OF EVERY REAL NUMBER IS EQUAL TO THE DISTANCE BETWEEN THIS
NUMBER AND THE COORDINATE ORIGIN ON THE NUMERICAL AXIS.
A symbol a is used for anotation of absolute value.
We can write findings from example 2 with using following equations (complete all empty spaces):
a) 1  1
d) 4 
g) 5,5 
b) 1  1
c)
4 4
e)
7
f)
7 
h) 5,5 
Now try to complete a definition of the absolute value:
THE DEFINITION OF THE ABSOLUTE VALUE a OF A REAL NUMBER a IS:
IF a  0 , THEN a  ……,
IF a  0 , THEN a  …… .
Absolute value of non-negative number a is equal to …. , absolute value of negative number a is egual
to a opposite number, which is written ……..
So for every real number we can notice:
|a| 0
(fill in the gap the right sign of inequality)
1
3. Calculate:
a) 105 
d) 7  5  8 
b) 15  (3) 
c)
29 
e)
2 2 
f)
1 2 
4. Mark on the numerical axis all numbers which are solutions of these equations.
a) x  5 __________________________________________________________________
b) x  5 __________________________________________________________________
c)
x  3 __________________________________________________________________
d) x  2 __________________________________________________________________
5. Calculate:
1 3 
a)
3 1 
b)
c)
46 
64 
Look at the previous exercise once more!
Each pair of equations have the same solutions. Is it possible that also these types of absolute value have
some geometrical interpretation?
Mark numbers 1 and 3 on the numerical axis:
Mark numbers 3 and 7 on the numerical axis:
What is the geometrical interpretation of the absolute value of the difference between numbers 1 and 3?
( 1  3 )?
…………………………………………………………………………………………(Write your idea.)
THE DISTANCE BETWEEN REAL NUMBERS a, b ON THE NUMERICAL AXIS IS EQUAL
TO …………. ,
or ……….….
6. Mark on the numerical axis all numbers which are solutions of these equations:
a) x 1  2
d) x  4  1
b) x  6  1
c)
x 1  5
e)
x  4 1
2