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Chapter 2: Problem 66

Evaluate. $$ |-x| \text { if } x=1 $$

Short Answer

Expert verified
|-x| when x=1 is 1.

Step by step solution

01

Understand Absolute Value

The absolute value of a number, written as \(|x|\), represents the distance from zero on the number line and is always non-negative. This means \(|-x| = |-1| = 1\) when \(x = 1\).
02

Substitution

Substitute \(x = 1\) into the expression \(|-x|\). We have: \(|-1|\).
03

Simplify the Absolute Value Expression

Compute the absolute value \(|-1|\). Since absolute value denotes the distance from zero, \(|-1| = 1\). Therefore, \(|-x| = 1\) when \(x = 1\).

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Key Concepts

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

Understanding the Number Line
When we talk about the absolute value of a number, we're dealing with its distance from zero on a number line.This concept is quite intuitive when you visualize it. Imagine a horizontal line with zero in the center and positive numbers extending to the right, while negative numbers extend to the left.
  • Zero is the starting point.
  • To the right of zero, numbers are positive.
  • To the left of zero, numbers are negative.
The absolute value, represented by vertical bars like e.g., \(|-x|\), important because it shows us the magnitude of a number regardless of its direction from zero on this number line.For instance,
\(|-1|\) is 1 because if you step in either direction from zero, the distance is the same.Thus, the absolute value always comes out as a non-negative number.
Mastering Substitution
Substitution is a powerful tool we often use to simplify expressions or equations.It involves replacing a variable in an expression with its actual value. This helps to make the expression more straightforward.In our example:
  • The expression is \(|-x|\).
  • We know that \(x=1\).
  • Hence, we substitute 1 into the expression.
After substitution, we replace \(x\) with 1 in the expression, which gives us \(|-1|\).This replacement step simplifies the analytical part of dealing with abstractions and brings us closer to the exact value or result.
Effective Simplification
Simplification is the process of reducing the expression to its most straightforward form.In mathematics, simplifying an expression can include a variety of operations, but most hinge on clarity and conciseness.Starting with our expression, which is now \(|-1|\), we analyze it:
  • The absolute value sign \((|.|)\) asks us to consider how far \(-1\) is from zero, which is 1.
  • This means \(|-1| = 1\).
Thus, we arrive at our simplified conclusion鈥擻(|-x| = 1\) when \(x = 1\).The essence of simplification lies in extracting the simplest form of a complex expression,making it easy to interpret and work with. This step-by-step process ensures clarity and precision in our calculations.

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