简化表达式

(Video) Evaluating Expressions Involving Absolute Value

by PatrickJMT

This video by Patrick JMT covers absolute value expressions.

$\large{\text{Summary}}$

Evaluating these expressions isn't much harder than regular expressions. Absolute value bars act like parentheses; you should do the operations inside them first.

He looks at three examples to show how to simplify them.

The first example is: $|8-12|$.

The first step is to do the operation inside the absolute value.

$$|8-12|=|-4|$$

Next, find the absolute value of the term.

$$|-4|=4$$

It is important to keep in mind that you cannot break apart the terms in the absolute value. $|8-12|\ne |8|-|12|$. We saw something similar when evaluating radical expressions.

The next example is $|-4||3|$.

We first take the absolute value of both terms.

$$|-4|=4\quad\text{and}\quad |3|=3$$

Next, multiply the terms together.

$$4\cdot3=12$$

The final example is: $|\frac{-25}{5}|$.

We start by dividing the terms inside.

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

Finally, find the absolute value of the term.

$$|-5|=5$$

The absolute value of a number is always positive. It measures the distance of a number or expression from $0$. For instance, $| 3 | = 3$, and $| –3 | = 3$ also.

Because we are measuring distance, we will always end up with a positive integer or a positive real number.

We can also see that $|x|=3$ could mean that $x=3$ or $x=-3$.

Let's take a look at what this concept would look like on a number line.

Let's take a look at some example problems.

Great! In this example, you basically found $|-4|$, because you found the distance from $-4$ to $0$. $$|-4|=4$$

Let's try another problem.

We can take absolute values one step further by evaluating absolute value equations, absolute value inequalities, and absolute value piecewise functions. If you want to dive even deeper, you can learn how to graph absolute value functions!

Absolute Value Expressions

Remember, when you take the absolute value of a number, the result is the non-negative version of that number. For example, $|12|=12,$ $|0|=0,$ and $|-12|=12.$

To simplify absolute value expressions, first use your knowledge of order of operations/PEMDAS to simplify the expressions within the absolute value signs. Then, take the necessary absolute values. Then simplify the whole expression!