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Chapter 2: Problem 32

In Exercises 15-32, express each set using the roster method. \(\\{x \mid x+3=9\\}\)

Short Answer

Expert verified
The set \(\{x \mid x+3=9\}\) expressed using the roster method is \{6\}.

Step by step solution

01

Identify the Given Set

The problem presents a set in set-builder notation: \(\{x \mid x+3=9\}\). In set-builder notation, the part before the vertical line '|' (here 'x') describes the members of the set, while the part following the vertical line provides the condition or rule that the members must fulfill. So here the set contains the values of 'x' that make the equation 'x + 3 = 9' true.
02

Solve the Equation

The next step is to solve the equation for 'x'. The equation \(x + 3 = 9\) is a simple linear equation. To solve for 'x', subtract 3 from both sides of the equation: \(x + 3 - 3 = 9 - 3\), which simplifies to \(x = 6\). So this is the solution of the equation.
03

Express in Roster Method

The final step is to express the set using the roster method, also known as the 'list' or 'tabular' method. In this method, we list out all the elements of the set within curly brackets '{ }'. Since there is only one solution to the equation, the set has just one element, 6. So the set, expressed using the roster method, becomes \{6\}.

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Key Concepts

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

Roster Method
The roster method is a straightforward way to represent sets by listing out all the elements in curly brackets. It's also known as the 'list' or 'tabular' method. When using the roster method, each element of the set is separated by a comma inside the curly brackets.
For example, the set containing all primary colors would be written as \[ \{ \text{red}, \text{blue}, \text{yellow} \} \].
In our exercise, after solving the equation, we found the solution to be 6. Therefore, when expressing this set using the roster method, it becomes \[ \{6\} \].
This method is user-friendly and particularly helpful when the sets are small or when you know all the members.
Set-Builder Notation
Set-builder notation is used to define sets by stating a property that its members must satisfy. It鈥檚 a concise way to describe the elements in the set without having to list them all.
In our exercise, the set is written as \( \{x \mid x+3=9\} \).
The part before the vertical line '|' shows the variable name (here 'x'), and the part after the line describes the condition or rule that x must satisfy.

This notation is particularly useful for defining large sets or sets with elements that follow a specific pattern.
  • The expression \( x+3=9 \) allows us to identify what x must be.
  • By solving the condition, we find that x equals 6.
Linear Equations
Linear equations are equations of the first degree, involving only the simplest operations and a variable raised to the power of one. They typically follow the format of \( ax + b = c \), where a, b, and c are constants.
Solving these equations involves isolating the variable on one side of the equation. In our specific problem, we have the equation \( x+3=9 \).
Here, the process involves:
  • Subtracting 3 from both sides to eliminate the constant from the left side.
  • This simplifies to \( x = 6 \), which is the solution.

The simplicity of linear equations makes them an excellent foundational topic in algebra, helping students build confidence for tackling more complex mathematical problems.

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Most popular questions from this chapter

Find each of the following sets. \(A \cap \varnothing\)

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