In this section, we will learn how to deal with absolute values.

 

Every real number can be represented by a point on the real number line.

The distance from a number $x$  on the real line to the origin (zero) is what we called the magnitude (weight) of that number in Section 1.1. Mathematically, this is called the absolute value of the number, and is denoted by $|x|$. 

For example, the distance from the point −5  to the origin is 5 units, so |-5| = 5.

 

Definition of the Absolute Value: 

  • $|x|=\begin{cases} -x & \text{ if } x<0\\ 0 & \text{ if } x=0\\ x & \text{ if } x>0\end{cases}$

Examples: 

a)  $| 7|  = 7$

b)   $|-7|  = 7$

c)   $\left|-\displaystyle \frac{1}{4}\right|  = \displaystyle \frac{1}{4}$

d)  $|-3| + |2| = 3 + 2 = 5$

e)  $|-3 + 2| = |-1| = 1$

f)  $|-7| – |-5| = 7 – 5 = 2$

g)  $-|7| – |-5| = -7 – 5 = -12$      

 

Practice Problems: 

(a)  $|-7| + 5$

(b)   $-|-7| + 5$

(c)   $|8| – |-4|$

(d)   $|-6|^2$

(e)   $|-15|\cdot |6|$

(f)   $\displaystyle\frac{|-21|}{|-7|}$

 

Answer Key: (a)   $12$     (b)    $-2$      (c)    $4$       (d)    $36$        (e)    $90$        (f)    $3$

 

For more detailed explanation, please read: Arithmetic|Algebra Appendix B.