/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 56 Simplify the expression. $$|-4... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 56

Simplify the expression. $$|-4|$$

Short Answer

Expert verified
|-4| = 4

Step by step solution

01

Understand Absolute Value

Absolute value represents the distance of a number from zero, regardless of its direction on the number line. It is always a non-negative number.
02

Apply the Definition

The absolute value of a number is the number itself if it's positive or zero, and the opposite of the number if it's negative. Here, the given number is (-4).
03

Calculate the Absolute Value

Since (-4) is negative, its absolute value is the opposite of (-4). Therefore: |-4| = 4.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Simplifying Expressions
Simplifying expressions is an essential skill in mathematics. It involves rewriting an expression in a simpler form without changing its value. To do this, one must follow certain mathematical rules and properties.

When simplifying absolute value expressions, remember that the absolute value of a number represents its distance from zero. For example, \(|-4|\) means we find the distance of -4 from zero. This distance is always given as a non-negative number.

Here is the process for simplifying \(|-4|\):
  • First, identify the number inside the absolute value symbol.
  • If the number is negative, like -4, take its opposite.
  • If it鈥檚 positive or zero, it remains unchanged.

So, \(|-4|\) simplifies to 4 because the distance between -4 and 0 on the number line is 4 units.
Number Line
A number line is a visual tool that helps to understand the position and distance of numbers. It is a straight horizontal line with numbers placed at intervals.

In the case of absolute value, the number line makes it easy to see why the absolute value is always non-negative.

Here鈥檚 how you can use a number line to find the absolute value of -4:
  • Draw a number line with zero at the center.
  • Locate -4 on the left side of zero.
  • Count the units from -4 to 0. You will count 4 units.

This visual representation confirms that \(|-4|\) equals 4 because it's 4 units away from zero, regardless of its direction.
Non-Negative Numbers
Non-negative numbers are numbers that are either positive or zero. They do not include any negative numbers. Absolute values are always non-negative.

When dealing with absolute values, you transform negative numbers into their positive counterparts. For example, \(|-4|\) gives you 4, which is non-negative.

Essential points to remember:
  • Absolute values measure distance, which cannot be negative.
  • The absolute value of a positive number is the number itself.
  • The absolute value of zero is zero.
  • The absolute value of a negative number is its opposite (positive).

This is why when you encounter an expression like \(|-4|\), you automatically know the result will be in the non-negative number category, simplifying it to 4.

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Most popular questions from this chapter

One of the most famous blizzards in the United States was the blizzard of \(1888 .\) In one part of Nebraska, the temperature plunged from \(40^{\circ} \mathrm{F}\) to \(-25^{\circ} \mathrm{F}\) in \(5 \mathrm{hr}\). What was the average change in temperature during this time?

Fill in the blank with \(<,>,\) or \(=\) $$-44 \square-54$$

Jeff's balance in his checking account was 2036 dollar at the beginning of the week. During the week, he wrote two checks, made two deposits, and made one ATM withdrawal. What is his ending balance? $$\begin{array}{l|r} \hline \text { Check } & -\$ 150 \\ \hline \text { Check } & -\$ 25 \\ \hline \text { Paycheck (deposit) } & \$ 480 \\ \hline \text { ATM } & -\$ 200 \\ \hline \text { Cash (deposit) } & \$ 80 \\ \hline \end{array}$$

Assume \(a>0\) (this means that \(a\) is positive) and \(b<0\) (this means that \(b\) is negative). Find the sign of each expression. $$a(-b)$$

Find the range. The range of a set of numbers is the difference between the highest value and the lowest value. That is, range = highest - lowest. Find two integers whose difference is \(-6 .\) Answers may vary.
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