Recall that the absolute value of a number represents the distance of that number from the origin.
Let's take a look at the number line below.
Since and are at the same distance from the origin, we have
In general, for any real number
We can use this fact to simplify expressions containing absolute value. Let's see an example.
Simplify the expression
Since we can rewrite our expression as follows:
$|-4x|=$
|
a
|
$-|4x|$ |
|
b
|
$16x$ |
|
c
|
$-4$ |
|
d
|
$|4x|$ |
|
e
|
$4$ |
$\left| -\dfrac{4z}{7} \right| =$
|
a
|
$- \dfrac{4}{7}$ |
|
b
|
$-\left| \dfrac{4z}{7} \right|$ |
|
c
|
$\left| \dfrac{4z}{7} \right|$ |
|
d
|
$\dfrac{4}{7}$ |
|
e
|
$\left| \dfrac{4}{7} \right|$ |
The absolute value operation is distributive with respect to multiplication. This means that the absolute value of a product equals the product of the absolute values.
To demonstrate, consider the following expression:
We can evaluate this expression in two separate ways:
- The first is to find the absolute value of the product. In doing this, we get
- The second is to find the product of the absolute values. In doing this, we get
In general
The absolute value operation is also distributive with respect to division. In general,
Simplify the expression
Firstly, since we can write our expression as
Then, we distribute the absolute value over the multiplication, and simplify:
$|5x|=$
|
a
|
$5$ |
|
b
|
$-5|x|$ |
|
c
|
$-5$ |
|
d
|
$5|x|$ |
|
e
|
$-|5x|$ |
Simplify the expression $|-11x|.$
|
a
|
$-11$ |
|
b
|
$11|x|$ |
|
c
|
$-|11x|$ |
|
d
|
$-11|x|$ |
|
e
|
$11$ |
Simplify the expression
Firstly, since we can write our expression as
Then, we distribute the absolute value over the division, and simplify:
$\left|-\dfrac{3}{2x}\right|=$
|
a
|
$6x$ |
|
b
|
$-\dfrac{3}{2|x|}$ |
|
c
|
$\dfrac{3}{2|x|}$ |
|
d
|
$\dfrac{3}{2}$ |
|
e
|
$-6x$ |
Simplify the expression $\left|-\dfrac{5x}{3y}\right|.$
|
a
|
$\dfrac{5}{3}$ |
|
b
|
$15xy$ |
|
c
|
$\dfrac{5|x|}{3|y|}$ |
|
d
|
$-15xy$ |
|
e
|
$-\dfrac{5|x|}{3|y|}$ |