Problem 14 - Elementary and Intermediate Algebra

Question

a. Suppose that when solving a linear equation, the variable drops out, and the result is \(7=-1 .\) What is the solution set? b. Suppose that when solving a linear inequality, the variable drops out, and the result is \(6 \leq 10 .\) Write the solution set in interval notation and graph it.

Step-by-Step Solution

Answer

a. Solution set: \(\emptyset\); b. Solution set: \((-\infty, \infty)\).

Step 1: Analyze the Linear Equation

In the linear equation resulting in \(7=-1\), notice that the statement is false. This means that no value for the variable could make the equation true, and therefore, the solution set is the empty set.

Step 2: Solution for Linear Equation

Since the equation leads to a false statement (like \(7=-1\)), the solution set for the equation is an empty set, represented as \(\emptyset\).

Step 3: Analyze the Linear Inequality

For the linear inequality resulting in \(6 \leq 10\), notice that the statement is true regardless of the value of the variable. This implies that any real number will satisfy the inequality.

Step 4: Solution in Interval Notation

Since the inequality \(6 \leq 10\) is always true, the solution set includes all real numbers. Thus, the solution in interval notation is \((-\infty, \infty)\).

Step 5: Graph the Inequality Solution Set

To graph \((-\infty, \infty)\), draw a number line and shade the entire line, indicating that all points on the line are included in the solution set.

Key Concepts

The Solution Set

In mathematical terms, the solution set is a collection of all the possible solutions that satisfy a given equation or inequality. When we solve equations or inequalities, we're trying to find values for the variable that make the statement true.

For example, if we have an equation where the final result is false, like in part a of our problem where it says that \(7 = -1\), it indicates that there are no values of the variable that could ever make the equation true. As a result, the solution set is said to be the empty set because there are no solutions available. This leads us into our next section on the concept of an empty set.

Understanding the Empty Set

An empty set, often represented by the symbol \(\emptyset\), is simply a set that contains no elements. In the context of solving equations, an empty set means there is no solution that will satisfy the equation.

For instance, in example a from our exercise, when the equation simplifies down to an untrue statement such as \(7 = -1\), there are absolutely no values that can satisfy the equation. This outcome leads to the conclusion that the solution set for such an equation is just the empty set.

• An empty set is also sometimes referred to as a null set.
• It's important to recognize and understand this concept, especially when you come across false equations.

Using Interval Notation

Interval notation is a way of writing the solution set of inequalities. It provides a compact way to express ranges of numbers that comprise the solution set.

In our exercise, for part b, when the inequality \(6 \leq 10\) is analyzed, it is clear that the inequality will hold true for any value of the variable. Mathematically, this means every real number satisfies the inequality. Thus, the solution set can be written in interval notation as \((-inity, inity)\).

• The round brackets \((\) and \()\) are used to indicate that infinity is not a number that can be included.
• Interval notation is a handy tool for quickly representing ranges without having to write cumbersome set builder notations.

Additionally, this solution set can be represented by shading the entire number line, indicating that all real numbers are part of the solution.