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Chapter 2: Problem 33

Evaluate the given expression for \(x=2, y=-3,\) and \(z=-4.\) $$|x-y|$$

Short Answer

Expert verified
The value is 5.

Step by step solution

01

- Understand the Expression

The given expression is \(|x - y|\), where \(x = 2\) and \(y = -3\). The vertical bars denote the absolute value, which means taking the positive value regardless of whether the result inside is positive or negative.
02

- Substitute Given Values

Substitute the provided values into the expression: \(x = 2\) and \(y = -3\). This results in \[|2 - (-3)|\].
03

- Simplify Inside the Absolute Value

Simplify inside the absolute value bars first. Calculate the subtraction: \(2 - (-3) = 2 + 3 = 5\).
04

- Apply the Absolute Value

Finally, take the absolute value of the result. Since \(|5| = 5\), the final answer is \5\.

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Key Concepts

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

Absolute Value
Absolute value is a fundamental concept in mathematics that measures the distance of a number from zero on the number line. It is always a non-negative value. When you see vertical bars around a number or an expression, such as \(|x|\), it means you are looking for the absolute value. This is crucial when dealing with expressions that can result in negative values, as it ensures the result is always positive. For example, the absolute value of both 5 and -5 is 5, because both are 5 units away from zero.
Substitution
Substitution is a technique where we replace variables with given values. It's a straightforward process but must be done carefully to avoid mistakes. In the exercise, we substitute the given values of \(x = 2\) and \(y = -3\) into the expression \( |x - y| \). This makes the expression \( |2 - (-3)| \), where each number replaces its corresponding variable. This step transforms the expression from a general form into a specific numerical form that we can then simplify.
Simplification
Simplification involves performing arithmetic operations to reduce an expression to its most straightforward form. After substitution, we had the expression \(2 - (-3) \). Simplifying inside the absolute value bars involves changing subtraction of a negative number into addition: \(2 - (-3) = 2 + 3 = 5\). Simplifying the expression this way ensures calculations are easier to handle and understand.
Positive Value
A positive value is any number greater than zero. In the context of absolute values, the result is always positive. In our exercise, after simplifying inside the absolute value to get \(5\), applying the absolute value does not change it because 5 is already positive. This ensures that regardless of the original values, the outcome is non-negative. Thus, \|5| = 5 \ , confirming the final result in our simplified expression.

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