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Chapter 3: Problem 21

Simplify each absolute value expression. (a) \(|7|\) (b) \(|-25|\) (c) \(|0|\)

Short Answer

Expert verified
a) 7, b) 25, c) 0

Step by step solution

01

Simplify \(|7|\)

The absolute value of a number represents its distance from zero on the number line, regardless of direction. Therefore, simplifying \(|7|\) simply means determining its distance from zero. Since 7 is 7 units away from zero on the number line, \(|7| = 7\).
02

Simplify \(|-25|\)

Similarly, to find the absolute value of \(|-25|\), determine its distance from zero. Since -25 is 25 units away from zero on the number line regardless of its negative sign, \(|-25| = 25\).
03

Simplify \(|0|\)

The absolute value of zero is itself because zero is at zero distance from zero. Therefore, \(|0| = 0\).

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Key Concepts

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

distance from zero
Understanding the absolute value of a number begins with comprehending the 'distance from zero' concept. When we talk about absolute value, we are referring to how far a number is from zero on the number line, without considering the direction.

For example:
  • The absolute value of 7, written as \(|7|\), is 7 because 7 is 7 units away from zero.
  • Likewise, the absolute value of -25, written as \(|-25|\), is 25 because -25 is 25 units away from zero. The negative sign does not affect the distance.
  • Lastly, the absolute value of 0, written as \(|0|\), is 0 because zero is already at zero distance from itself.
By understanding the concept of 'distance from zero,' you will always be able to find the absolute value of any number.
number line
A number line is a visual representation of numbers arranged in ascending order on a straight line. It helps in understanding the concept of absolute value better.

Here's how the number line assists in finding absolute values:
  • Positive numbers: When you move right from zero, you encounter positive numbers like 1, 2, 3, etc. The distance between these numbers and zero is their absolute value. For example, 7 is 7 units to the right of zero, so \(|7| = 7|\).
  • Negative numbers: Moving left from zero gives you negative numbers like -1, -2, -3, etc. Despite the negative sign, the distance from zero is measured in positive units. For instance, -25 is 25 units to the left of zero, thus \(|-25| = 25|\).
  • Zero: Zero is a unique point on the number line because it is the reference point. Zero's distance from itself is zero, hence \(|0| = 0|\).
Visualizing numbers on a number line makes understanding and simplifying absolute values much easier.
simplification
To simplify an absolute value expression means to find its numerical value without considering its sign.

The process is straightforward:
  • Take the given number inside the absolute value symbols (\(| |\)).
  • Identify its distance from zero on the number line.
  • Convert this distance into a positive number, which is the simplified absolute value.
Let's revisit the examples:
  • For \(|7|\), since 7 is already positive, its absolute value is 7.
  • For \(|-25|\), although -25 is negative, we consider its distance from zero, which is 25 units. Hence, \(|-25| = 25\).
  • For \(|0|\), zero is at zero distance from itself, so \(|0| = 0\).
Simplifying absolute values is crucial for effortlessly solving more complex mathematical problems involving absolute values.

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Most popular questions from this chapter

In the following exercises, determine whether each number is a solution of the given equation. $$ 5 x-10=-35 $$ $$ \begin{array}{l}{\text { (a) } x=-9} \\ {\text { (b) } x=-5 \quad \text { (c) } x=5}\end{array} $$

In the following exercises, translate to an algebraic expression and simplify if possible. the product of \(-12\) and 6

In the following exercises, solve. Early one morning, the temperature in Syracuse was \(-8^{\circ} \mathrm{F}\) . By noon, it had risen \(12^{\circ} .\) What was the temperature at noon?

In the following exercises, evaluate each expression. $$ 16-2 n \text { when } n=-8 $$

In the following exercises, divide. $$ 96 \div(-16) $$
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