Problem 7 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

List the elements of the set in roster notation. $$ \\{x \mid 2-x=4 \text { and } x \text { is an integer }\\} $$

Step-by-Step Solution

Answer

The set in roster notation is \(\{2\}\).

Step 1: Solve the equation

First, let's solve the equation to find the integer value of \(x\): $$ 2-x=4 $$ To solve for \(x\), we need to isolate it on one side of the equation. We can do this by adding \(x\) to both sides and subtracting 4 from both sides: $$ 2+(-x)=4 \\ 2-4=-x \\ -2=-x $$ Now we need to divide both sides by -1 to get the value of \(x\): $$ x=(-2)/(-1) $$ Finally, we find that \(x=2\).

Step 2: Write the set in roster notation

We have found that the only integer value of \(x\) satisfying the condition is 2. Now we can write the set in roster notation. $$ \\{x \mid 2-x=4 \text { and } x \text { is an integer }\\} = \{2\} $$ So, the set contains only one element, which is \(\{2\}\).

Key Concepts

Equation Solving

Equation solving is a fundamental part of mathematics where you aim to find the value of the unknown variable that makes the equation true. Here, we start with a simple linear equation, \(2 - x = 4\), where our goal is to find the value of \(x\). To solve it, we perform several steps to "isolate" the variable \(x\), getting it by itself on one side of the equation.

• First, we can manipulate the equation by adding \(x\) to both sides to remove the negative sign in front of \(x\). This results in \(2 = 4 + x\).
• Next, we rearrange to solve for \(x\) by subtracting 4 from both sides. This gives us \(2 - 4 = x\), which simplifies to \(-2 = x\).
• Finally, to express \(x\) positively, multiply both sides by -1, yielding \(x = 2\).

Thus, solving equations typically involves logical manipulations like addition, subtraction, division, or multiplication to find the solution.

Integer Solutions

Finding integer solutions means that we are interested in solutions to equations that are whole numbers, including negatives, zero, and positives. For our problem, the equation \(2 - x = 4\) must produce an integer value when solved for \(x\).

• Once we solve the equation and get \(x = 2\), we check that \(x\) is indeed an integer. Integers are not fractions or decimals, so numbers like 2 fit perfectly as they are whole.
• It’s crucial to confirm the integer requirement when working through problems like this. Here, our solution \(2\) satisfies the conditions of the problem because it seamlessly fits into the set of integers.
• This attention to the type of solution (integer, real number, etc.) helps ensure that your solution aligns with the problem's stipulations, particularly in contexts demanding specific types, like integers.

Set Notation

Set notation is a mathematical language used to describe a collection of objects, known as elements. When we express solutions in set notation, we convey which specific elements satisfy a particular condition.
For our exercise, after finding the integer solution to the equation \(2-x=4\), we represent this in set notation. Here we’re using roster notation, which is a way of listing all elements of a set within curly brackets.

• The original problem asks us to form a set where the elements satisfy the equation and are integers. Since the solution is \(x = 2\), we write it down as \(\{2\}\).
• Roster notation is straightforward: if there's only one solution, like here, we just list that single element inside the brackets. Multiple solutions would simply be separated by commas, e.g., \(\{2, 3, 4\}\).

This way of expressing sets ensures clarity and communicates precisely which value or values meet the equation's criteria. Understanding this representation is key in various fields that use mathematics.