Problem 79 - Elementary Algebra

Question

Solve and graph the solution set. In addition, present the solution set in interval notation. $$ 12 x-2<-1 \text { or } 12 x-2>1 $$

Step-by-Step Solution

Answer

The solution set is \((-\infty, \frac{1}{12}) \cup (\frac{1}{4}, \infty)\).

Step 1: Solve the first inequality

The first inequality is \(12x - 2 < -1\). Add 2 to both sides to isolate the term with \(x\): \[12x < 1\] Next, divide both sides by 12:\[x < \frac{1}{12}\]

Step 2: Solve the second inequality

The second inequality is \(12x - 2 > 1\). Add 2 to both sides to simplify:\[12x > 3\] Then divide both sides by 12:\[x > \frac{1}{4}\]

Step 3: Determine the solution set

Since these are 'or' inequalities, the solution set is any \(x\) that satisfies either \(x < \frac{1}{12}\) or \(x > \frac{1}{4}\).

Step 4: Express in interval notation

Represent the solution set in interval notation. Given \(x < \frac{1}{12}\) or \(x > \frac{1}{4}\), the intervals are:\[(-\infty, \frac{1}{12}) \cup (\frac{1}{4}, \infty)\]

Step 5: Graph the solution set

To graph the intervals \((-\infty, \frac{1}{12})\) and \((\frac{1}{4}, \infty)\), draw two rays on a number line which extend from negative infinity to \(\frac{1}{12}\) (with an open circle at \(\frac{1}{12}\)) and from \(\frac{1}{4}\) to positive infinity (with an open circle at \(\frac{1}{4}\)).

Key Concepts

Graphing Inequalities

Graphing inequalities involves visualizing solution sets on a coordinate plane or line. Here, we focus on a number line. To graph an inequality like \(x < \frac{1}{12}\) or \(x > \frac{1}{4}\), use open circles and rays. Open circles are used for \("<"\) or \(">"\) to indicate that the endpoint isn't included in the solution.

For \(x < \frac{1}{12}\), place an open circle at \(\frac{1}{12}\) and draw a ray extending leftwards, indicating all values less than \(\frac{1}{12}\).

For \(x > \frac{1}{4}\), place an open circle at \(\frac{1}{4}\) and draw a ray extending rightwards, indicating all values greater than \(\frac{1}{4}\).

This visual representation makes it easier to understand which values satisfy the inequality.

Interval Notation

Interval notation is a concise way to describe a set of values that satisfy an inequality. It uses parentheses and brackets to denote "from" and "to" values along with inclusivity.

For example, in our problem, the solution is \(x < \frac{1}{12}\) or \(x > \frac{1}{4}\). Using interval notation, represent this as:

• \((-\infty, \frac{1}{12})\) for \(x < \frac{1}{12}\)
• \((\frac{1}{4}, \infty)\) for \(x > \frac{1}{4}\)

These intervals are combined with the union symbol (\(\cup\)) to indicate the solution includes either set of values. Parentheses are used here because infinity is not a specific number and cannot be included.

Number Line Representation

A number line is a simple but effective tool for representing inequalities. To depict our solution, draw a horizontal line with significant numbers marked for reference, such as \(\frac{1}{12}\) and \(\frac{1}{4}\).

Place an open circle at both \(\frac{1}{12}\) and \(\frac{1}{4}\) to represent exclusion. Then, visualize:

• A leftward-pointing ray starting from \(\frac{1}{12}\) for values \(x < \frac{1}{12}\)
• A rightward-pointing ray beginning from \(\frac{1}{4}\) for values \(x > \frac{1}{4}\)

This illustration clearly shows when either inequality holds true, providing insight into which intervals satisfy the conditions set by the equations.