Problem 13 - College Algebra with Modeling and Visualization

Question

Solve the inequality symbolically. Express the solution set in set-builder or interval notation. $$ \frac{t+2}{3} \geq 5 $$

Step-by-Step Solution

Answer

The solution set is \([13, \infty)\).

Step 1: Multiply Both Sides by 3

To eliminate the fraction, multiply both sides of the inequality by 3: \[ 3 \cdot \frac{t+2}{3} \geq 3 \cdot 5 \]. This simplifies to \( t + 2 \geq 15 \).

Step 2: Subtract 2 from Both Sides

Subtract 2 from both sides of the inequality to isolate \( t \): \[ t + 2 - 2 \geq 15 - 2 \]. This gives \( t \geq 13 \).

Step 3: Write the Solution in Interval Notation

The solution \( t \geq 13 \) means \( t \) is greater than or equal to 13. In interval notation, this is expressed as \([13, \, \infty)\).

Step 4: Verify Solution

To verify, choose a value for \( t \) from the interval, such as 14, and substitute it back into the original inequality: \( \frac{14+2}{3} \geq 5 \), which simplifies to \( \frac{16}{3} \geq 5 \) or approximately \( 5.33 \geq 5 \). This confirms the solution is correct.

Key Concepts

Set-builder Notation

Set-builder notation is a mathematical language used to describe a set by detailing the properties that members of the set must satisfy. It's a concise way to specify which elements are included in a set, particularly useful for describing an infinite set. This notation sets the stage for precisely defining complicated sets, especially when dealing with inequalities.
For example, the solution set for the inequality \( t \geq 13 \) can be expressed in set-builder notation as:

• \( \{ t \mid t \geq 13 \} \)

This reads as "the set of all \( t \) such that \( t \) is greater than or equal to 13."
Set-builder notation effectively communicates the characteristics of the set without listing every single number within the solution.

Interval Notation

Interval notation is a method of describing sets of numbers that fall within a particular range. Unlike set-builder notation, it focuses strictly on the endpoints and whether those endpoints are included in the set.
For inequalities such as \( t \geq 13 \), interval notation offers a streamlined expression:

• The closed bracket, \([ \), signifies that 13 is included in the solution.
• The infinity symbol, \( \infty \), indicates that the interval continues indefinitely to the right.
• The parenthesis, \( ) \), signifies that infinity is not a definitive endpoint and cannot be "included."

Thus, the solution is written as \([13, \infty)\), succinctly capturing the range of numbers without needing verbal or lengthy text explanations. This notation is particularly helpful in calculus and higher-level math, where precision and brevity are essential.

Verification of Solutions

Verification is a crucial step in solving inequalities, confirming that the calculated solutions meet the necessary conditions. By substituting a value from the solution set back into the original inequality, we validate its correctness.
Consider \( t \geq 13 \): to verify, choose a value, say \( t = 14 \), within the solution set:\

• Substitute \( t = 14 \) back into the original inequality: \[ \frac{14+2}{3} \geq 5 \]
• Simplify to obtain \( \frac{16}{3} \geq 5 \), which evaluates to approximately \( 5.33 \geq 5 \).

Since \( 5.33 \) is indeed greater than \( 5 \), the value verifies the solution.
Verifying solutions ensures no arithmetic mistakes were made in solving or interpreting the inequality. It adds confidence in the accuracy of results and prevents misunderstandings that may arise from overlooking this simple but essential step.