Problem 33 - Algebra and Trigonometry

Question

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

Interval: (5/2, 11/2].

Step 1: Break the Compound Inequality Into Two Separate Inequalities

The given compound inequality is \( \frac{2}{3} \geq \frac{2x-3}{12} > \frac{1}{6} \). This can be split into two separate inequalities: \( \frac{2}{3} \geq \frac{2x-3}{12} \) and \( \frac{2x-3}{12} > \frac{1}{6} \). We will solve each inequality individually.

Step 2: Solve the First Inequality

Starting with \( \frac{2}{3} \geq \frac{2x-3}{12} \), multiply both sides by 12 to eliminate the fraction on the right: \[ 12 \times \frac{2}{3} \geq 2x-3 \]This simplifies to:\[ 8 \geq 2x-3 \]. Add 3 to both sides to get:\[ 11 \geq 2x \].Finally, divide by 2: \[ \frac{11}{2} \geq x \], or \( x \leq \frac{11}{2} \).

Step 3: Solve the Second Inequality

Now solve \( \frac{2x-3}{12} > \frac{1}{6} \). Multiply both sides by 12 to clear the fraction:\[ 2x-3 > 2 \]. Add 3 to both sides:\[ 2x > 5 \].Now, divide by 2:\[ x > \frac{5}{2} \].

Step 4: Combine the Solutions

To find \( x \) that satisfies both inequalities, combine the solutions: \( x \leq \frac{11}{2} \) and \( x > \frac{5}{2} \). The intersection of these inequalities is:\( \frac{5}{2} < x \leq \frac{11}{2} \).

Step 5: Express the Solution in Interval Notation

The solution \( \frac{5}{2} < x \leq \frac{11}{2} \) in interval notation is \((\frac{5}{2}, \frac{11}{2}]\).

Step 6: Graph the Solution Set

Draw a number line. Place an open circle at \( \frac{5}{2} \) because it is not included in the interval, and a closed dot at \( \frac{11}{2} \) because it is included. Shade the region between them to indicate all numbers in \((\frac{5}{2}, \frac{11}{2}]\) are part of the solution.

Key Concepts

Compound Inequality

A compound inequality involves combining two individual inequalities into one statement, usually by using conjunctions like "and" or "or". In our exercise, the compound inequality is \( \frac{2}{3} \geq \frac{2x-3}{12} > \frac{1}{6} \). Here, the phrase "and" is implied, meaning both conditions must be satisfied simultaneously. Thus:

• The first part \( \frac{2}{3} \geq \frac{2x-3}{12} \) demands that \( x \) satisfies this inequality.
• The second part \( \frac{2x-3}{12} > \frac{1}{6} \) must also hold true for the same \( x \) values.

To tackle such compound inequalities, break them down into two separate inequalities, solve them independently, and find the intersection of their solution sets. This ensures that both conditions are met by the values of \( x \). By individually solving each part, as shown in the original solution, we come to a shared solution set of \( \frac{5}{2} < x \leq \frac{11}{2} \). This means all \( x \) values within this range make both inequalities true.

Interval Notation

Interval notation is a concise way of representing a set of numbers, particularly useful when describing solutions to inequalities. It tells you the range of values that fulfill an inequality and specifies whether the endpoints are included or excluded through the use of parentheses \(()\) and brackets \([]\).

• Parentheses \(()\) signify that an endpoint is not included.
• Brackets \([]\) indicate that an endpoint is included.

For the solution \( \frac{5}{2} < x \leq \frac{11}{2} \), interval notation conveys this as \((\frac{5}{2}, \frac{11}{2}]\). Here, \( \frac{5}{2} \) has a parenthesis because it is not part of the solution, while \( \frac{11}{2} \) uses a bracket, showing inclusion. This notation provides a clean and clear way to describe the range of possible values without listing them individually.

Solution Set Graphing

Graphing the solution set of an inequality gives a visual representation that complements the algebraic solution. It's a helpful way to illustrate the span of the solution visually on a number line. Here's how to graph the given solution set \((\frac{5}{2}, \frac{11}{2}]\):

• First, draw a horizontal number line to represent all possible values of \( x \).
• Place an open circle at \( \frac{5}{2} \) because it is not included in the solution. Open circles are used to denote values that are not part of the set.
• Place a closed dot at \( \frac{11}{2} \). Closed dots mean the number is part of the solution.
• Shade the region between \( \frac{5}{2} \) and \( \frac{11}{2} \) along the number line. This shaded area represents every number between these limits that satisfies the inequality.

This graphical method quickly and effectively shows what numbers meet the inequality criteria, offering another layer of understanding beyond the algebraic notation.