Problem 33 - College Algebra with Corequisite Support

Question

For the following exercises, graph the function. Observe the points of intersection and shade the \(x\) -axis representing the solution set to the inequality. Show your graph and write your final answer in interval notation. $$ |x-1|>2 $$

Step-by-Step Solution

Answer

The solution is \((-\infty, -1) \cup (3, \infty)\).

Step 1: Understanding the Absolute Value Inequality

The inequality given is \(|x-1| > 2\). An absolute value inequality of the form \(|x-a|>b\) can be rewritten as two separate inequalities: \(x-a > b\) and \(x-a < -b\). Here, it becomes \(x-1>2\) or \(x-1<-2\).

Step 2: Separate into Two Inequalities

Solve the two inequalities obtained from the absolute value expression. Start with \(x-1>2\), which simplifies to \(x>3\). Second, solve \(x-1<-2\), which simplifies to \(x<-1\).

Step 3: Graph the Solutions on a Number Line

On a number line, graph the solution to the inequalities. For \(x>3\), shade the region to the right of 3. For \(x<-1\), shade the region to the left of -1. These two regions represent the values of \(x\) that satisfy \(|x-1|>2\).

Step 4: Write the Solution in Interval Notation

Combine the two regions to express the solution set in interval notation. The interval notation for \(x<-1\) is \((-\infty, -1)\) and for \(x>3\) is \((3, \infty)\). Therefore, the final solution is written as the union of the two intervals: \((-\infty, -1) \cup (3, \infty)\).

Key Concepts

Interval Notation

Interval notation is a concise way to describe a set of numbers or solutions to an inequality. It uses parentheses and brackets to show intervals on the number line.

• Parentheses, \( ...\), mean that an endpoint is not included, also referred to as 'open interval'.
• Brackets, \[ ...\], mean that an endpoint is included, known as a 'closed interval'.

For inequalities like \(|x-1| > 2\), once we determine that the solutions are for \(x < -1\) and \(x > 3\), we express this in interval notation to clearly convey the solution set. Specifics in this case:

• The interval for \(x < -1\) is written as \((-\infty, -1)\).
• The interval for \(x > 3\) is \( (3, \infty) \).

The union of these intervals, \((-\infty, -1) \cup (3, \infty)\), efficiently communicates that any number less than \(-1\) or greater than \(3\) satisfies the inequality.

Graphing Inequalities

Graphing inequalities on a number line is a visual way to represent solution sets. It's especially helpful when dealing with more complex inequalities involving absolute values.
To graph the inequality \(|x-1|>2\), we break it down into two parts:

• The inequality \(x-1>2\) implies that our solutions include all values greater than \(3\).
• The inequality \(x-1<-2\) tells us that values less than \(-1\) are part of the solution.

When graphing:

• Open dots or circles are used at numbers \(-1\) and \(3\), showing they're not included.
• Arrows or shading indicate the direction of the inequality, extending to negative infinity for \(x<-1\), and positive infinity for \(x>3\).

These graphical elements make it easy to visualize that the solution includes all numbers less than \(-1\) or greater than \(3\), aligning perfectly with our interval notation.

Number Line Representation

A number line is a straightforward representation of real numbers aligned in order from smallest to largest. It is a powerful visual aid, particularly when explaining inequalities.
The key steps for drawing on a number line for our inequality \(|x-1|>2\) include:

• Locate the points \(-1\) and \(3\), which are the boundaries derived from solving the separate inequalities.
• Plot these points with open circles to indicate they are not part of the solution set.
• Shade the line extending left from \(-1\) to show numbers less than \(-1\).
• Similarly, shade the line extending right from \(3\) for numbers greater than \(3\).

This number line clearly portrays the solution set, enhancing understanding by providing a quick glance at where solutions to the inequality lie.