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Chapter 0: Problem 57

Find each absolute value and simplify if needed. $$ |-9|-|-3| $$

Short Answer

Expert verified
6

Step by step solution

01

- Understand Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of direction. In other words, it is always non-negative.
02

- Find the Absolute Value of Each Number

Calculate the absolute value of -9 and -3. \(|-9| = 9\)and \(|-3| = 3\)
03

- Perform the Subtraction

Subtract the absolute value of -3 from the absolute value of -9:\(9 - 3\)
04

- Simplify the Result

Simplify the subtraction: \(9 - 3 = 6\)

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Key Concepts

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

Absolute Value
Absolute value represents a number's distance from zero on a number line, regardless of its direction. It is always a non-negative value.
For example, the absolute value of -9, which is \(|-9|\), is 9 because -9 is 9 units away from zero.
Similarly, the absolute value of -3 is 3. Essentially, \(|x|\) gives you the non-negative magnitude of any number x.
Number Line
A number line is a visual way to represent numbers; it includes positive, negative numbers, and zero.
On a number line, each point corresponds to a number. Zero is typically placed in the center.
Numbers to the right of zero are positive, and numbers to the left are negative.
For example, to find \(|-9|\), locate -9 on the number line and count its distance to zero, which is 9 units.
Non-Negative Numbers
Non-negative numbers are all numbers that are either zero or positive. They do not include any negative values. When we talk about absolute values, we are referring to non-negative numbers.
For instance, the absolute value of -9 is 9, and 9 is a non-negative number.
Similarly, the absolute value of -3 is 3, which is also non-negative.
Subtraction
Subtraction is a basic arithmetic operation used to find the difference between two numbers.
When working with absolute values, subtraction follows the same rules. For example, in the problem \(|-9| - |-3|\), you first find the absolute values (9 and 3), and then subtract 3 from 9.
The result is 6. This gives us the simplified form after subtraction. Subtraction of absolute values ensures we are always dealing with non-negative differences.

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Most popular questions from this chapter

Frank's grandson was asked to evaluate the following expression. \(9+15 \div 3\) Frank gave the answer as \(8,\) but his grandson gave the answer as \(14 .\) The grandson explained that the answer is 14 because of the "Order of Process rule," which says that when evaluating expressions, we proceed from right to left rather than left to right. (Note: This is a true story.) (a) Whose answer was correct for this expression, Frank's or his grandson's? (b) Was the reasoning for the correct answer valid? Explain.

Use an inequality symbol to write each statement. $$ 13-3 \text { is less than or equal to } 10 . $$

The distributive property can be used to mentally perform calculations. $$\begin{array}{l}38 \cdot 17+38 \cdot 3 \\\=38(17+3) \\\=38(20) \\\=760\end{array} $$Use the distributive property to calculate each value mentally. $$ 27 \cdot 60+27 \cdot 40 $$

Evaluate each expression for \(a=-3, b=64,\) and \(c=6 .\) $$ \frac{\sqrt{b}-4 a}{c^{2}} $$

Use an inequality symbol to write each statement. $$ 6+7 \text { is not equal to }-13 $$
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