Understanding Absolute Value in Mathematics

14 Questions

What is the absolute value of -9?

If x is a negative number, what is the absolute value of x?

-x

For any number x, what is the property |x| = -x true for?

x < 0

When solving the equation |2y + 4| = 10, what is the possible value of y?

In algebraic expressions, the absolute value can be applied to any term within an expression. For example, what is the absolute value of |-5 * 7|?

When finding the absolute value of a number, what happens if the number is zero?

It remains the same

What is the result of the expression |-2 * -3|?

When can you distribute the absolute value to each term within an expression in algebra?

Always, regardless of the signs of the terms

For which scenario would the absolute value of a number be undefined?

When the number is imaginary

Given the inequality |x| < 5, what are the possible values of x?

-5 < x < 5

If |m - 3| = 7, what are the potential solutions for m?

m = -4 or m = 10

For which values of x does |2x - 1| ≤ 3 hold true?

-2 ≤ x ≤ 1

If |5n + 2| > 8, what are the possible ranges for n?

-10 < n or n > 2

What is the solution set to the inequality |3y - 1| ≥ 4?

-1/3 ≤ y ≤ 5/3

Study Notes

Absolute Value: A Foundational Concept

Imagine you're trying to find a friend's address, but they only give you the distance from their house to a nearby landmark, either positive or negative. You'd still be able to plan your route, but you'd need to know the magnitude of that distance – not its direction – to do so accurately. In mathematics, the concept of the absolute value performs a similar function, providing the magnitude of a number without considering its sign.

Defining Absolute Value

The absolute value of a number, denoted as (|x|), is the non-negative value of the number itself. In simpler terms, it's the distance between the number and zero, regardless of whether the number is positive or negative.

[ |x| = \begin{cases} x, & \text{if } x \geq 0 \ -x, & \text{if } x < 0 \end{cases} ]

For example:

(|3| = 3)
(|-4| = 4)
(|0| = 0)

Properties of Absolute Value

The absolute value function has several important properties:

(|x| \geq 0) for all (x).
(|x| = x) for (x \geq 0) and (|x| = -x) for (x < 0).
The absolute value of a sum or difference does not change the sign of the terms: (|x + y| = |x| + |y|).
The absolute value of a product does not change the sign of the factors: (|xy| = |x| |y|).

Absolute Value and Equations

In solving equations, absolute values are often used to represent the possible values for a variable, where only the magnitude matters, not the sign. For example, consider the equation (|x - 3| = 5). The solution for this equation is (x = 8) and (x = -1), because both distances from (3) to (8) and from (3) to (-1) are (5).

Applications of Absolute Value

Absolute value has numerous applications, including distance calculations, graph analysis, and even in communication, as demonstrated by the growing use of the #no_search feature in search engines like Bing Chat, which allows users to request information without searching the web.

In summary, the absolute value function is a fundamental concept in mathematics that provides the magnitude of a number, regardless of its sign, and has numerous applications in various fields.