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Chapter 2: Problem 83

Fill in the blank with \(<,>,\) or \(=\) $$-|32| \square|0|$$

Short Answer

Expert verified
-32 < 0

Step by step solution

01

Find the absolute value

Calculate the absolute value of the numbers inside the absolute value symbols. For 32, the absolute value is \(32\), and for 0, the absolute value is \(0\).
02

Apply the negative sign

After calculating the absolute value, apply the negative sign to the \( |32| \) which gives \( -|32| = -32 \).
03

Compare the values

Now compare the values \( -32 \) and \( 0 \). Since -32 is less than 0, the inequality sign to use is \( < \).

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Key Concepts

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

absolute value
Absolute value is a way to show the distance a number is from zero, ignoring which direction. For any number, the absolute value is always a positive number or zero.
For example:
  • The absolute value of 32 is written as \(|32|\), which equals 32.
  • The absolute value of -32 is written as \(|-32|\), which also equals 32.
The absolute value function strips the negative sign (if there is one) and just returns the magnitude of the number. So, when you see \(|a|\), you just think about how far \(a\) is from zero.
inequalities
Inequalities are expressions that compare two values or quantities using symbols like \(<\), \(>\), or \(=\). Here is what each symbol means:
  • \(<\): Less than
  • \(>\): Greater than
  • \(=\): Equal to
For instance, consider the inequality \(5 < 10\). This means that 5 is less than 10.
Inequalities are essential in mathematics because they help us understand the relationship between numbers. Comparing values using inequalities can be applied in various scenarios, such as in our exercise, comparing negative numbers and zero.
negative numbers
Negative numbers are numbers less than zero, represented with a minus sign (-). For example, -32 is a negative number.
When it comes to comparing negative numbers or negative and positive numbers, keep in mind these simple rules:
  • Negative numbers are always less than positive numbers. For example, -5 is less than 3.
  • Among negative numbers, a smaller negative number is considered greater. For example, -2 is greater than -5.
In our exercise, turning the absolute value of 32 back into a negative form (-32) makes it easier to compare with zero. Since negative numbers are less than zero, -32 is less than zero, so \(-32 < 0\).

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Most popular questions from this chapter

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

Simplify. $$(-5)^{4}$$

Divide the real numbers, if possible. $$\frac{300}{-2}$$

Perform the indicated operation. $$18(-6)$$

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