Absolute Value

The Absolute Value of a Number

of a number refers to its distance from zero on the number line. It is denoted by a vertical line (bar) on each side of the number. For example, "the absolute value of -5" is written as |-5|.

5 and -5 on the number line. Since the two numbers are located at the same distance from 0, they have the same absolute value

As you can see, $\hspace{0.2em} 5 \hspace{0.2em}$ and $\hspace{0.2em} -5 \hspace{0.2em}$ both lie at the same distance from $\hspace{0.2em} 0 \hspace{0.2em}$ , and so they have the same absolute value (positive $\hspace{0.2em} 5 \hspace{0.2em}$ ).

|5| = |{-5}| = 5

This is true for all numbers and their negatives.

So to get the absolute value of a number, just drop its negative sign (if it has one).

Now, let's look at some example problems based on this concept.

How to Simplify Absolute Value Expressions | Examples

Example

Simplify -

(a) \hspace{0.3cm} |4|

(b) \hspace{0.3cm} |-12|

Solution (a)

The absolute value of a positive number is the number itself. So,

|4| = 4

Solution (b)

To get the absolute value of a negative number, we get rid of the negative sign. So,

|{-12}| = 12

It's that simple.

Example

Simplify -

(a) \hspace{0.3cm} |{-2} \times 3|

(b) \hspace{0.3cm} |{-5} + 8|

Solution (a)

To get the absolute value of an expression, it's important to first simplify it.

So here, we multiply the two numbers together and then take the absolute value of the product.

\begin{align*} |{-2} \times 3| &= |{-6}| \\[1em] &= 6 \end{align*}

Solution (b)

Again, first, we simplify the expression inside the bars.

\begin{align*} |{-5} + 8| &= |3| \\[1em] &= 3 \end{align*}

Example

Simplify -

(a) \hspace{0.3cm} -|{-7}|

(b) \hspace{0.3cm} -|4 \times 10|

Solution (a)

- {\color{Red} |{-7}|} = - {\color{Red} 7}

Through this example, I wanted to highlight that the negative sign outside the bars remains unaffected. We take the absolute value of what's inside them.

Solution (b)

Same story with this one. The external negative sign remains unaffected.

\begin{align*} - {\color{Red} |4 \times 10|} &= - {\color{Red} |40|} \\[1em] &= - {\color{Red} 40} \end{align*}

Difference of Two Numbers

Example

Simplify -

(a) \hspace{0.3cm} |3 - 5|

(b) \hspace{0.3cm} |5 - 3|

Solution (a)

\begin{align*} |3 - 5| &= |{-2}| \\[1em] &= 2 \end{align*}

Solution (b)

\begin{align*} |5 - 3| &= |2| \\[1em] &= 2 \end{align*}

See how in both the examples, we got the same result? So,

|3 - 5| = |5 - 3|

This is true for any two numbers.

The order in which we subtract one number from another becomes irrelevant when we take the absolute value of the difference. In general,

|a-b|=|b-a|

And with that, we come to the end of this elementary tutorial on absolute value. Until next time.