Problem 8 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

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

Step-by-Step Solution

Answer

The set in roster notation is an empty set: \(\{\}\).

Step 1: Solve the equation: \(2-x=4\)

To solve the equation, we need to isolate \(x\). We can do this by adding \(x\) to both sides of the equation and subtracting \(4\) from both sides: \(2 - x + x = 4 + x\) \(2 = 4 + x\) Now, subtract 4: \(2 - 4 = 4 - 4 + x\) \(-2 = x\) The solution of the given equation is \(x = -2\). Step 2: Check if the solution is a fraction

Step 2: Check if x is a fraction

The solution we found is -2, which is an integer. Therefore, the given conditions are not met since the found value of \(x\) is not a fraction. Step 3: Write the set in roster notation

Step 3: Set in roster notation

Since there are no values of x that satisfy both conditions (solving the equation and being a fraction), the set will be empty. We represent an empty set in roster notation as: \(\{\}\) So the answer is an empty set: $$ \{x \mid 2-x=4 \text { and } x \text { is a fraction }\} = \{\} $$

Key Concepts

Roster Notation

Roster notation is a simple and visual way to define a set by listing its elements. When using roster notation, you enclose the listed items within curly braces \(\{\} \), and separate them with commas if there's more than one element. For example, the set of all vowels can be written as \(\{a, e, i, o, u\}\). This form of notation is straightforward, making it easy to see exactly what elements are part of the set.

In some situations, such as when solving equations, you may end up with an empty set. This happens when no solutions satisfy all given conditions. In roster notation, an empty set is represented simply as \(\{\} \), indicating that there are no elements satisfying the set definition.

In our example problem, we needed a value for \(x\) that satisfies the equation and is also a fraction. Since no such value exists, our set is empty, shown in roster notation as \(\{\} \). This reflects that there are no fractions meeting the defined criteria.

Solving Equations

Solving equations is a fundamental skill in mathematics. It involves finding the value(s) of the variable(s) that make the equation true. To solve an equation, you often perform operations like addition, subtraction, multiplication, or division to isolate the variable.

In our exercise, the equation given was \(2 - x = 4\). The goal was to find the value of \(x\). Here are the steps we followed:

• First, add \(x\) to both sides. This eliminates the negative sign in front of \(x\), giving us \(2 = 4 + x\).
• Next, subtract 4 from both sides to isolate \(x\). This results in \(2 - 4 = x\) or \(-2 = x\).

The solution we found is \(x = -2\). However, this solution does not meet the additional requirement of being a fraction, as it's an integer.

Fractions

Fractions are numbers that represent parts of a whole, expressed as a ratio of two integers: the numerator (top part) and the denominator (bottom part). They are written in the form \(\frac{a}{b}\) where \(a\) is the numerator and \(b\) is the denominator, and \(b eq 0\).

Fractions can represent values between whole numbers as well as values greater than one or less than zero. They are especially useful for expressing exact divisions, like cutting a pizza into 8 equal slices, having \(\frac{1}{8}\) of it.

In our problem, after solving the equation, we checked if the solution \(x = -2\) is a fraction. Since \(-2\) is an integer, it is not a fraction because it can be expressed without a denominator other than 1, i.e., \(-2 = \frac{-2}{1}\). Therefore, it does not satisfy the condition of being a fraction, requiring us to reconsider the solution in the context of a set problem.