Problem 14 - College Algebra with Modeling and Visualization

Question

Solve the inequality symbolically. Express the solution set in set-builder or interval notation. $$ \frac{2-t}{6}<0 $$

Step-by-Step Solution

Answer

\((2, \infty)\)

Step 1: Understand the inequality

We have the inequality \( \frac{2-t}{6} < 0 \). It involves a fraction that needs to be analyzed to determine the range of values for \( t \) that make this inequality true.

Step 2: Set the numerator less than zero

The term \( \frac{2-t}{6} \) will be negative if its numerator \( 2 - t \) is negative (since the denominator 6 is positive). Therefore, we set the inequality \( 2 - t < 0 \).

Step 3: Solve for t

To solve \( 2 - t < 0 \), we rearrange the terms. This can be done by subtracting 2 from both sides to get \(-t < -2\). Next, we multiply both sides by -1, remembering to reverse the inequality sign, giving us \( t > 2 \).

Step 4: Express the solution in interval notation

The solution from the previous steps tells us that \( t \) must be greater than 2. In interval notation, this is expressed as \((2, \infty)\), indicating all numbers greater than 2.

Key Concepts

Symbolic Manipulation

Symbolic manipulation involves working with mathematical symbols to rearrange and simplify expressions. In the context of solving inequalities, this means adjusting the inequality to isolate the variable of interest.

When faced with the inequality \( \frac{2-t}{6} < 0 \), the first step in symbolic manipulation is understanding that this inequality must be simplified to solve for \( t \).

The term \( \frac{2-t}{6} \) is already somewhat simplified as a fraction. However, focusing on the numerator simplifies our task. Thus, we recognize that the inequality holds if \( 2 - t < 0 \), since a negative numerator will result in the fraction being negative given the positive nature of the denominator.

Symbolic manipulation in this context helps us understand and solve the inequality without having to directly compute values in the denominator until after rearranging terms.

Interval Notation

Interval notation is a compact way of representing a range of values, often used in expressing the solution to inequalities.

Once the inequality \( 2-t < 0 \) is manipulated and solved for \( t \), resulting in \( t > 2 \), we use interval notation to record this.

In interval notation, a parenthesis \( ( ) \) is used to denote that the endpoint value is not included, whereas a bracket \( [ ] \) indicates inclusion. Thus, \( (2, \infty) \) represents all real numbers greater than 2, but not including 2 itself.

Using interval notation is efficient and clear, especially when dealing with complex ranges or conveying concise mathematical solutions.

Numerator Analysis

Numerator analysis is crucial when solving inequalities involving fractions. The sign of the fraction is influenced significantly by the numerator.

In the expression \( \frac{2-t}{6} < 0 \), the constant denominator \( 6 \) does not change the sign. The critical analysis lies in the numerator \( 2 - t \).

To determine when the fraction is negative, we analyze the condition \( 2 - t < 0 \).

This simplifies to \( t > 2 \), indicating the numerator becomes negative when \( t \) becomes greater than 2.

Understanding numerator analysis helps in pinpointing where the inequality flips its state, allowing one to solve inequalities involving fractions more effectively.