Question

Which of these expresses an equivalent relationship? \begin{tabular}{l|l|l} (A) \(|-3|\) & \(=-|3|\) & OR \end{tabular} (B) \(|3|=|-3|\)

Short answer

Option (B): \\(|3| = |-3|\\) is the equivalent relationship.

Step-by-step solution

Understanding Absolute Value

The absolute value of a number, denoted as \(|x|\), represents its distance from zero on the number line, regardless of direction. This means absolute value is always non-negative.

Calculate Each Expression

Evaluate the expressions given in the options: \(|-3|\) evaluates to 3 because \(-3\) is 3 units away from zero.Similarly, \(|3|\) evaluates to 3 because 3 is 3 units from zero.So, \(|-3| = 3\) and \(|3| = 3\).

Analyze Option (A)

For option (A): \(|-3| = -|3|\) simplifies to \(|-3| = -3\). From the previous evaluation, \(|-3| = 3\). Therefore, \(-|3| = -3\), which means \(|-3|\) and \(-|3|\) are not equal, so this relationship is not equivalent.

Analyze Option (B)

For option (B): \(|3| = |-3|\). Both expressions evaluate to 3, as determined earlier. Therefore, this relationship is equal and maintains the \ equivalent \ expression.

Key concepts

Equivalent Expressions

In mathematics, equivalent expressions are expressions that, when simplified, yield the same result. It means they hold true for all values that they are defined for. For example,

• expression 1 and expression 2 are equivalent if their evaluations lead to the same value under the same conditions.
• In the original exercise, \( |3| \) and \( |-3| \) are equivalent expressions because both simplify to \( 3 \).

The absolute value plays a crucial role here, showing that the distance or magnitude, irrespective of direction, remains identical for both positive and negative instances of a number. Understanding equivalent expressions helps in solving equations and simplifies mathematical computations. It is an essential skill tested in various exams, including the PSAT mathematics section.

Number Line

A number line is a visual representation of numbers arranged sequentially. It helps to understand numerical relationships and operations, such as addition and subtraction, by visualizing them spatially. Let's explore some properties:

• Numbers increase from left to right on the number line.
• Each point on the line represents a real number corresponding to its distance from zero.
• This tool is particularly useful to demonstrate absolute value, where a number's distance from zero is what matters, not its direction.

In the exercise, both \(|-3|\) and \(|3|\) refer to points located \(3\) units away from zero on the number line, showing how they are indeed equivalent in terms of absolute value. This visual understanding ensures clarity in dealing with expressions and comprehension of absolute value concepts.

Non-negative

The term 'non-negative' in mathematics refers to numbers that are either positive or zero. Absolute values are non-negative because they measure distance, which cannot be less than zero:

• The absolute value of any real number, positive or negative, results in a non-negative number.
• For instance, \(|3|\) and \(|-3|\) both yield a result of \(3\), which is non-negative.

This concept underscores why expressions involving absolute values are often simplified into non-negative equivalents, irrespective of the direction on the number line. Remember, absolute values ignore direction, focusing solely on magnitude.

PSAT Mathematics

The PSAT mathematics section tests your understanding of various mathematical principles, including algebraic concepts, geometry, and data analysis. Absolute value is a commonly tested topic. Here’s why:

• It assesses your capability to interpret numerical relationships and make logical connections.
• Questions might involve simplifying expressions or solving equations where absolute values are present.
• Recognizing equivalent expressions and understanding non-negative results is vital.

By mastering these absolute value principles, you are better equipped for success in the PSAT mathematics section. Grasping how expressions relate through absolute values can ease your way through slightly more complex problems.