Problem 4 - College Algebra with Corequisite Support

Question

When solving an inequality, we arrive at: $$\begin{array}{c} x+2>x+3 \\ 2>3 \end{array}$$ Explain what our solution set is.

Step-by-Step Solution

Answer

The solution set is the empty set \(\emptyset\), as the inequality simplifies to a false statement.

Step 1: Understanding the Inequality

The inequality given is \(x + 2 > x + 3\). Our goal here is to determine the values of \(x\) that satisfy this inequality.

Step 2: Simply the Inequality

Subtract \(x\) from both sides of the equation to simplify it:\[x + 2 - x > x + 3 - x \2 > 3\]This simplifies to \(2 > 3\).

Step 3: Interpreting the Result

The statement \(2 > 3\) is false because 2 is not greater than 3. Therefore, the original inequality has no solution as the expression simplifies to a false statement.

Step 4: Conclusion

Since the inequality simplifies to an always false statement, there are no values of \(x\) that can satisfy the inequality. Hence, the solution set is the empty set, denoted as \(\emptyset\).

Key Concepts

Inequality Simplification

When working with inequalities, one of the first steps is simplification. Simplifying an inequality means making it as straightforward as possible without changing its meaning.

To simplify the inequality, we perform similar operations as we do with equations, such as adding or subtracting the same number from both sides. This process maintains the inequality's balance and makes it easier to solve.

In our example, we start with the inequality:

• \(x + 2 > x + 3\)

By subtracting \(x\) from both sides, we remove the \(x\) term and get:

• \(2 > 3\)

This step is crucial for reducing the inequality to its simplest form before making further evaluations. Remember, always check if additional operations could clarify the inequality without altering its core.

False Statements in Inequalities

Sometimes, simplifying an inequality leads us to a statement that doesn't hold true. A false statement means no values will satisfy the inequality.

In our example, after simplification, we arrived at:

• \(2 > 3\)

This is clearly false, as 2 is not greater than 3. When an inequality simplifies to such a statement, it indicates a critical understanding: The original problem has no solution.

A false statement in inequalities provides a definitive end to the search for any valid solutions, because an incorrect comparison shows that no number, including variables or constants, will satisfy the condition of the inequality.

Always ensure to recognize these moments, as they directly impact the determination of the solution set and final conclusions.

Solution Sets in Algebra

The solution set of an inequality consists of all the values that satisfy it. In algebra, determining the solution set is a crucial aspect as it tells us the range or specific values that make an inequality true.

However, when an inequality reduces to a false statement, like \(2 > 3\), the solution set is empty. This means there are no possible values for \(x\) to make the inequality true.

• The notation for an empty solution set is \(\emptyset\).
• The solution set can also be described as having no elements.

This is because nothing fits into the condition set by our simplified inequality.

Understanding solution sets is significant because it describes all potential solutions in a precise format. Learning how to interpret when a solution set is empty helps clarify scenarios where constraints cannot be satisfied.