Problem 25 - College Algebra with Modeling and Visualization

Question

Solve the inequality symbolically. Express the solution set in set-builder or interval notation. $$ 5<4 t-1 \leq 11 $$

Step-by-Step Solution

Answer

The solution in interval notation is \( \left( \frac{3}{2}, 3 \right] \).

Step 1: Break the Compound Inequality

The compound inequality is given as \( 5 < 4t - 1 \leq 11 \). We can break it into two separate inequalities: \( 5 < 4t - 1 \) and \( 4t - 1 \leq 11 \). We will solve each inequality separately.

Step 2: Solve the First Inequality

First, solve \( 5 < 4t - 1 \). Add 1 to both sides to get \( 6 < 4t \). Then, divide both sides by 4 to isolate \( t \), resulting in \( t > \frac{3}{2} \).

Step 3: Solve the Second Inequality

Next, solve \( 4t - 1 \leq 11 \). Add 1 to both sides to get \( 4t \leq 12 \). Then divide both sides by 4 to isolate \( t \), giving \( t \leq 3 \).

Step 4: Combine the Solutions

The solution from Step 2 is \( t > \frac{3}{2} \) and from Step 3 is \( t \leq 3 \). Combining these gives the solution \( \frac{3}{2} < t \leq 3 \).

Step 5: Write the Solution in Interval Notation

The solution \( \frac{3}{2} < t \leq 3 \) can be expressed in interval notation as \( \left( \frac{3}{2}, 3 \right] \).

Key Concepts

Understanding Compound Inequalities

Compound inequalities are mathematical expressions that combine two distinct inequalities into one statement. These inequalities involve an 'and' or 'or', denoting different relationships between the variables. In this case, the inequality \(5 < 4t - 1 \leq 11\) uses 'and' implicitly. This implies that both conditions should be satisfied simultaneously. To solve a compound inequality, often it is divided into two separate inequalities.

• \(5 < 4t - 1\)
• \(4t - 1 \le 11\)

Each side of the compound inequality is dealt with separately to determine the values of \(t\) that satisfy both conditions. After solving each inequality, the results are combined to find the range of values that satisfy the entire compound inequality.

Solution Set Essentials

Once each part of the compound inequality is solved, we need to find the solution set. A solution set is essentially a collection of all possible solutions that satisfy the given inequality. It signifies where both inequalities intersect on a number line.
In our exercise:

• Solving \( 5 < 4t - 1 \) gives \( t > \frac{3}{2} \)
• Solving \( 4t - 1 \leq 11 \) gives \( t \leq 3 \)

The combined solution set for this compound inequality is \( \frac{3}{2} < t \leq 3 \). This means that all values of \(t\) within this range satisfy both parts of the compound inequality. Solution sets help us understand the limits within which a variable must lie to satisfy conditions given by inequalities.

Expressing in Interval Notation

Interval notation is a streamlined way of representing a set of numbers, showing the beginning and end of an inequality's solution. For a compound inequality, it offers a concise expression of the accepted range of values.
To write interval notation:

• Use parentheses \(()\) to denote that an endpoint is not included (open interval).
• Use brackets \([]\) to show that an endpoint is included (closed interval).

For the compound inequality in our exercise, \(\frac{3}{2} < t \leq 3\), the interval notation is \(\left(\frac{3}{2}, 3\right]\). This reflects that \(t\) is greater than \(\frac{3}{2}\) (but not equal to) and less than or equal to 3. Each type of interval notation helps in quickly understanding and communicating the range of solutions visually and symbolically.