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Chapter 0: Problem 62

In Exercises \(61-66,\) evaluate each algebraic expression for \(x=2\) and \(y=-5\) $$|x-y|$$

Short Answer

Expert verified
The evaluation of \(|x-y|\) for \(x=2\) and \(y=-5\) is \(7\).

Step by step solution

01

Substitute

Firstly, substitute the given values of \(x\) and \(y\) into the expression \(\|x - y\|\) to get the absolute difference. So the expression becomes \(\|2 - (-5)\|\).
02

Simplify Inside Absolute Value

Simplify inside the absolute value symbol. The expression \(2 - (-5)\) equals \(2 + 5\), therefore, the operation inside the absolute value marker becomes \(7\).
03

Evaluate Absolute Value

Since the absolute value of a positive number is the number itself, the absolute value of \(7\) is \(7\).

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Key Concepts

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

Understanding Algebraic Expressions
Algebraic expressions are mathematical phrases that can include numbers, variables, and operations. They help us represent real-world scenarios in a mathematical form. For example, in our exercise, the expression \(|x - y|\) uses variables \(x\) and \(y\) along with subtraction and the absolute value operation.

  • Variables are symbols, like \(x\) and \(y\), that represent unknown or changeable values. They allow us to generalize problems and find solutions under different conditions.
  • Operations like addition, subtraction, multiplication, and division indicate how variables and numbers are related to each other.
  • The absolute value operation, denoted by \(| |\), shows the distance of a number from zero on the number line, always yielding a non-negative result.
Recognizing how these components come together is essential for solving algebraic expressions. Mastering the basics of algebraic expressions paved the way for tackling more complex equations and problems.
The Role of Substitution in Algebra
Substitution is a powerful technique used to evaluate algebraic expressions by replacing variables with given numerical values. It simplifies expressions by temporarily removing variables.

  • Identify the values that need to substitute the variables. In our exercise, \(x\) becomes \(2\) and \(y\) becomes \(-5\).
  • Replace each variable in the expression with its corresponding value. This transforms the expression \(|x - y|\) to \(|2 - (-5)|\).
  • Perform any arithmetic operations after substitution to simplify the expression further, which brings us to: \(|2 + 5|\).
Substitution makes it easier for you to work with abstract mathematical phrases by converting them into simple numerical calculations. By practicing substitution, you can solve various algebraic problems efficiently.
Simplifying Expressions and Calculating Absolute Values
Simplification transforms algebraic expressions into their simplest form, making calculations easier. Here are some key steps from the exercise that demonstrate simplification:
  • After substitution, simplify the arithmetic operation within the absolute value symbols. For this exercise, simplify \(2 + 5\) to get \(7\).
  • Evaluate the absolute value of the simplified expression. Since \(7\) is already positive, \(|7|\) equals \(7\).

Understanding the absolute value function is crucial as it outputs the distance from zero on the number line. This means whether a number is positive or negative, its absolute value is positive or zero.

By practicing simplification, you can resolve expressions more quickly and handle more complex algebraic problems with confidence. The goal is to reach an expression that's as straightforward and easy to understand as possible.

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Most popular questions from this chapter

List all numbers that must be excluded from the domain of each rational expression. $$\frac{3}{2 x^{2}+4 x-9}$$

The formula for converting Fahrenheit temperature, \(F,\) to Celsius temperature, \(C\), is $$C=\frac{5}{9}(F-32)$$ If Celsius temperature ranges from \(15^{\circ}\) to \(35^{\circ},\) inclusive, what is the range for the Fahrenheit temperature? Use interval notation to express this range.

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The equation \((2 x-3)^{2}=25\) is equivalent to \(2 x-3=5\)

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Although I can solve \(3 x+\frac{1}{5}=\frac{1}{4}\) by first subtracting \(\frac{1}{5}\) from both sides, I find it easier to begin by multiplying both sides by \(20,\) the least common denominator.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. In an inequality such as \(5 x+4<8 x-5,\) I can avoid division by a negative number depending on which side I collect the variable terms and on which side I collect the constant terms.
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