Problem 31 - Algebra and Trigonometry

Question

\(9-32\) me solve the linear inequality. Express the solution using interval notation and graph the solution set. $$ \frac{2}{3} \geq \frac{2 x-3}{12}>\frac{1}{6} $$

Step-by-Step Solution

Answer

The solution is \( \left( \frac{5}{2}, \frac{11}{2} \right] \).

Step 1: Understanding the compound inequality

We have the compound inequality \( \frac{2}{3} \geq \frac{2x-3}{12} > \frac{1}{6} \). This means we need to solve two inequalities: \( \frac{2}{3} \geq \frac{2x-3}{12} \) and \( \frac{2x-3}{12} > \frac{1}{6} \). We will handle each part separately.

Step 2: Solve the inequality \( \frac{2x-3}{12} \leq \frac{2}{3} \)

First, solve \( \frac{2x-3}{12} \leq \frac{2}{3} \). Multiply both sides by 12 to remove the fraction: \( 2x - 3 \leq 8 \). Add 3 to both sides: \( 2x \leq 11 \). Finally, divide both sides by 2: \( x \leq \frac{11}{2} \).

Step 3: Solve the inequality \( \frac{2x-3}{12} > \frac{1}{6} \)

Next, solve \( \frac{2x-3}{12} > \frac{1}{6} \). Multiply both sides by 12: \( 2x - 3 > 2 \). Add 3 to both sides: \( 2x > 5 \). Finally, divide both sides by 2: \( x > \frac{5}{2} \).

Step 4: Combine and express the solution in interval notation

Combine the results of the two inequalities: \( \frac{5}{2} < x \leq \frac{11}{2} \). This solution means \( x \) is greater than \( \frac{5}{2} \) and less than or equal to \( \frac{11}{2} \). In interval notation, the solution is \( \left( \frac{5}{2}, \frac{11}{2} \right] \).

Step 5: Graph the solution set

To graph the solution set \( \left( \frac{5}{2}, \frac{11}{2} \right] \), draw a number line and mark an open circle at \( \frac{5}{2} \) to indicate that this endpoint is not included, and a closed circle at \( \frac{11}{2} \) to indicate the endpoint is included. Shade the region between the two circles to represent all numbers between \( \frac{5}{2} \) and \( \frac{11}{2} \).

Key Concepts

Compound Inequalities

When you come across a compound inequality, you're essentially dealing with more than one inequality problem at once. A compound inequality combines two inequalities that are connected by the words "and" or "or." Let's break this down:

• An "and" compound inequality requires both conditions to be true at the same time. For example, in the inequality \( \frac{2}{3} \geq \frac{2x-3}{12} > \frac{1}{6} \), we need the expression \( \frac{2x-3}{12} \) to satisfy both \( \frac{2}{3} \geq \frac{2x-3}{12} \) and \( \frac{2x-3}{12} > \frac{1}{6} \).
• An "or" compound inequality, on the other hand, requires only one of the conditions to be true. This isn't our case here, but this helps understand how compound inequalities can vary.

Start by solving each inequality separately, just like in any other linear inequality problem. Then, you combine the solutions. It's the combination that gives compound inequalities their unique feature, representing a range or interval of numbers rather than just a single value. This concept allows for more complex relationships between expressions and increases our flexibility in representing real-world constraints.

Interval Notation

Interval notation is a simple way of writing down the set of numbers (or solution set) that are part of a solution to an inequality. Once you've solved your inequalities, interval notation helps express your answers succinctly. How does it work?

• The symbol \((\) or \()\) is used to indicate that an endpoint is not included in the set (open interval).
• Similarly, \([\) or \(]\) signifies that the endpoint is included (closed interval).

For the solution \( \frac{5}{2} < x \leq \frac{11}{2} \), interval notation is \(( \frac{5}{2}, \frac{11}{2}]\). Notice how we use "(" for \( \frac{5}{2} \) since it's not part of the solution, and "]" for \( \frac{11}{2} \) since it is included.
This notation is great because:

• It's concise and clean.
• It quickly communicates which parts of the number line are included.
• It's widely used in mathematics and makes reading solutions easier.

Graphing Inequalities

Graphing can provide a visual representation that complements interval notation. A graph shows exactly which numbers make the inequality true on a number line. Let's explore how this works:

• Start with a number line. Identify the relevant points (here, \( \frac{5}{2} \) and \( \frac{11}{2} \)).
• Use an open circle at \( \frac{5}{2} \), because this point is not included in the solution set. This signifies that the value \( x \) strictly cannot be \( \frac{5}{2} \).
• Place a closed circle at \( \frac{11}{2} \) since it is part of the solution set, meaning \( x \) can equal \( \frac{11}{2} \).
• Shade the region between these points to indicate all numbers within this range are part of the solution.

Visual graphs help in understanding the relationship between numbers or expressions, providing clarification beyond numerical and algebraic expressions. It's a vital tool for homework, visuals in exams, or real-world applications where seeing the range is equally important as knowing it numerically.