Simplifying Absolute Value Expressions

The absolute value of a number refers to its distance from 0 on the number line. It's represented as |a|, which defines the magnitude of any integer 'a'. The absolute value of any integer, whether it is positive or negative, will be a real number, no matter which sign it has. The two vertical lines that represent absolute value are known as modulus symbols. An absolute value is always a positive number and never a negative number.

Examples of absolute value numbers

Whether the number in the modulus symbols is positive or negative, the absolute value is always positive.

Example 1

The absolute value of $\left|3\right|=3$

The absolute value of $|-3|=3$

The absolute value of decimals works the same way as that of whole numbers.

Example 2

The absolute value of $\left|387.291\right|=387.291$

The absolute value of $|-387.291|=387.291$

The same is true of absolute numbers that are fractions.

Example 3

The absolute value of $\left|\frac{1}{3}\right|=\frac{1}{3}$

The absolute value of $|-\frac{1}{3}|=\frac{1}{3}$

And as you would expect, the same is true for mixed numbers.

The absolute value of $\left|34\frac{2}{9}\right|=34\frac{2}{9}$

The absolute value of $\left|-34\frac{2}{9}\right|=34\frac{2}{9}$

It's not actually necessary to write "the absolute value of $\left|x\right|$ " because that is what $\left|x\right|$ means. So an absolute value statement can be written as:

Example 4

$\left|x^2\right|=x^2$

$|-x^2|=x^2$

Writing multiplication expressions with absolute values

It is possible to write expressions using absolute values. For example:

$\left|-4\right|\times \left|3\right|=\left|12\right|$

It would be the same with any combination of positive and negative symbols using the same numbers, because what the expression is really saying is:

$4\times 3=12$

It's also possible to write expressions within the absolute value symbols, such as:

$\left|8(-4)\right|$

To simplify this expression, you need to know that the absolute value of a product is the same as the product of the absolute values.

So

$|8(-4)|=\left|8\right||-4|=\left(8\right)\left(4\right)=32$

Here are a few more examples of the same concept.

$|(-12)(-8)|=|-12||-8|=12\times 4=32$

Sometimes you can drop absolute value signs on variables.

$\left|x^2{y}^3\right|=\left|x^2\right|\left|y^3\right|$

$\left|x^2{y}^3\right|=x^2\left|y^3\right|$

Since any real value squared is positive we can drop the absolute value signs around $x^2$ but not $y^3$ .

Writing division expressions with absolute values

The same rule applies to quotients in division expressions. The absolute value of a quotient is the quotient of the absolute values.

$\left|\frac{-8}{4}\right|=\frac{\left|8\right|}{|-4|}=\frac{8}{4}=2$

Here are a few more examples of the same concept:

$\left|\frac{-9}{3}\right|=\frac{\left|9\right|}{|-3|}=\frac{9}{3}=3$

$-\left|\frac{-15}{6}\right|=\frac{-\left|-15\right|}{\left|6\right|}=\frac{-15}{6}=-2.5$

Writing addition and subtraction expressions with absolute values

The same rule does not apply for addition and subtraction expressions with absolute values.

Because you perform the addition or subtraction within the modulus symbols before simplifying the expression, there is a difference between $\left|-a+b\right|$ and $\left|a\right|+\left|b\right|$ .

For example,

$\left|-5+8\right|=\left|3\right|=3$

but

$\left|-5\right|+\left|8\right|=5+8=13$

However,

$\left|5+8\right|=5+8=13$

and

$\left|5\right|+\left|8\right|=5+8=13$

do end up with the same solution. It's necessary to be careful and pay close attention to the negative signs when performing addition or subtraction in expressions with absolute values.

Practice simplifying absolute value expressions

a. Simplify the following expression:

$\left|\left(12\right)(-5)\right|$

First, separate the factors into two separate absolute value statements.

$\left|12\right||-5|$

Then rewrite the problem using the absolute values of each number.

$12\times 5$

Then solve the expression.

$12\times 5=60$

b. Simplify the following expression:

$\left|(-17)(-10)\right|$

First, separate the numbers into two absolute value statements.

$|-17||-10|$

Then write the expression using the absolute values.

$17\times 10$

Finally, simplify by performing the multiplication.

$17\times 10=170$

c. Simplify the following expression:

$\left|-a^5{b}^3\right|$

Simplify by removing the negative and separating the expressions into separate absolute value statements.

$\left|a^5\right|\left|b^3\right|$

d. Simplify the following expression:

$-|-\frac{25}{5}|$

Separate the numbers into separate absolute value statements.

$-\frac{|-25|}{\left|5\right|}$

Then write the expression using the absolute values.

$-\frac{25}{5}$

Finally, simplify by performing the division.

$-5$

e. Simplify the following expression:

$\left|\frac{-28}{8}\right|$

Separate the number into an absolute value statement.

$\frac{\left|-28\right|}{\left|8\right|}$

Then write the expression using the absolute values.

$\frac{28}{8}$

Finally, simplify by performing the division.

$\frac{28}{8}=3.5$

f. Simplify the following expression:

$\left|15+17\right|$

Separate the numbers into separate absolute value statements.

$\left|15\right|+\left|17\right|$

Finally, simplify by performing the addition.

$15+17=32$

g. Simplify the following expression:

$\left||-|9+8|\right|$

Simplify by separating the expression into separate absolute value statements.

$\left||-|1|\right|$

The absolute value of -1 is 1.

h. Simplify the following expression:

$\left|-\left|-9\right|+\left|8\right|\right|$

First, calculate the absolute values separately.

$\left|-9\right|+\left|8\right|$

Finally, simplify by performing the addition.

$9+8=17$

Topics related to the Simplifying Absolute Value Expressions

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Absolute Value Functions

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Get help learning about simplifying absolute value expressions

Private tutoring is an excellent way for your student to learn about absolute value and specifically how to simplify absolute value expressions. It can be challenging to keep track of the negatives and positives when working with absolute values and even the best math student might want a little extra help in memorizing the concepts and rules they need to keep in mind when working with absolute values.

A tutor will work with your student at their pace, taking the extra time necessary when they are struggling with a particular concept and spending less time on material that your student picks up quickly. A tutor can also learn your student's learning style and customize their lessons to take advantage of it, making lessons effective and efficient. By complementing their in-school lessons, a tutor's assistance can help your student develop the understanding they need.

A tutor is there with your student as they work on problems, guiding them in the right way to do them from the get-go, so your student doesn't develop bad math habits. Your student can ask questions as they arise, giving them every opportunity to correctly perform the calculations necessary to simplify absolute value expressions from the start. Tutoring is beneficial to all types of students, whether they are struggling or whether they want to learn advanced concepts.

To learn more about how tutoring can benefit your student, contact the Educational Directors at Varsity Tutors today. We look forward to talking with you and finding you a qualified tutor.