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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

Simplify each expression. $$ -(-x-y) $$

When simplifying an expression, we usually do some steps mentally.Providing the property that justifies each statement in the given simplification. $$3 x+4+2 x+7$$ $$ =3 x+(2 x+4)+7 $$

Perform the indicated operations $$ -9-4-(-3)+6 $$

Simplify each expression. $$ 0.4(10-5 x)-0.8(5+10 x) $$

Sea level refers to the surface of the ocean. The depth of a body of water can be expressed as a negative number, representing average depth in feet below sea level. The altitude of a mountain can be expressed as a positive number, indicating its height in feet above sea level. The table gives selected depths and altitudes. $$ \begin{array}{|l|c|l|c|} \hline \text { Body of Water } & \begin{array}{c} \text { Average Depth in Feet } \\ \text { (as a negative number) } \end{array} & \text { Mountain } & \begin{array}{c} \text { Altitude in Feet } \\ \text { (as a positive number) } \end{array} \\ \hline \text { Pacific Ocean } & -14,040 & \text { Denali } & 20,310 \\ \text { South China Sea } & -4802 & \text { Point Success } & 14,164 \\ \text { Gulf of California } & -2375 & \text { Matlalcueyetl } & 14,636 \\ \text { Caribbean Sea } & -8448 & \text { Rainier } & 14,410 \\ \text { Indian Ocean } & -12,800 & \text { Steele } & 16,624 \\ \hline \end{array} $$ List the mountains in order, starting with the shortest and ending with the tallest.
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