Problem 64 - Elementary Algebra

Question

Solve and graph the solution set. In addition, present the solution set in interval notation. $$ -12<34 x-23 \leq 12 $$

Step-by-Step Solution

Answer

The solution set is \(\left(\frac{11}{34}, \frac{35}{34}\right]\).

Step 1: Understanding the Inequalities

The inequality to solve is given as \(-12 < 34x - 23 \leq 12\). It is a compound inequality, meaning it includes two separate inequalities that must be satisfied simultaneously.

Step 2: Solving the First Inequality

Start with the inequality \(-12 < 34x - 23\). To isolate \(x\), first add 23 to both sides: \(-12 + 23 < 34x\). This simplifies to \(11 < 34x\). Then divide both sides by 34: \(\frac{11}{34} < x\).

Step 3: Solving the Second Inequality

Now solve the inequality \(34x - 23 \leq 12\). First, add 23 to both sides: \(34x \leq 35\). Then divide both sides by 34: \(x \leq \frac{35}{34}\).

Step 4: Combining the Solution

The solutions \(\frac{11}{34} < x\) and \(x \leq \frac{35}{34}\) can be combined. The solution set in interval notation is \(\left(\frac{11}{34}, \frac{35}{34}\right]\).

Step 5: Graphing the Solution Set

To graph the solution set, draw a number line. Represent \(\frac{11}{34}\) with an open circle, indicating \(x\) is greater than \(\frac{11}{34}\). Represent \(\frac{35}{34}\) with a closed circle, indicating \(x\) is less than or equal to \(\frac{35}{34}\). Shade the region between these points.

Key Concepts

Inequality Solving

When faced with inequality solving, it's important to understand that you're finding the range of values that the variable can take to keep the inequality true. Let's break down the given compound inequality:

• First, we had \(-12 < 34x - 23\). This is the first inequality we need to solve.
• Next, we use simple algebra: Add 23 to each side to get \(11 < 34x\).
• Finally, divide both sides by 34: \(\frac{11}{34} < x\).

For the second inequality \(34x - 23 \leq 12\):

• Start by adding 23 to both sides: \(34x \leq 35\).
• Then divide by 34 to isolate \(x\), giving \(x \leq \frac{35}{34}\).

The solution is the range where both inequalities are true. It is a systematic approach where we solve each part separately and then merge the results.

Interval Notation

Once you solve inequalities, interval notation is a concise way to express the solution set. Here are some basics:

• An open bracket \( ( \) indicates that an endpoint is not included in the interval, meaning the inequality is 'less than' or 'greater than'.
• A closed bracket \( ] \) means the endpoint is included, expressing 'less than or equal to' or 'greater than or equal to'.
• The interval \(\left(\frac{11}{34}, \frac{35}{34}\right]\) means that \(x\) is greater than \(\frac{11}{34}\) but less than or equal to \(\frac{35}{34}\).

Interval notation is efficient. It provides a visual cue of the values \(x\) can assume.

Graphing Inequalities

Graphing inequalities on a number line is a visual method to show solutions.

• Start by drawing a basic number line.
• Place an open circle at \(\frac{11}{34}\) to indicate that this is where the solution starts, but this point isn't included.
• Put a closed circle at \(\frac{35}{34}\), showing this point is included in the solution.
• Shade the space between these points. This highlighted section shows all possible values of \(x\) that fulfill the compound inequality.

This graphical approach makes it easier to understand the practical range of solutions. It's visually intuitive and immediately highlights the differences between strict and inclusive inequality limits.