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Chapter 2: Problem 33

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. $$ |x-1|>2 $$

Short Answer

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

Step by step solution

01

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 \(x-1<-2\).
02

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 \(x<-1\).
03

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 \(|x-1|>2\).
04

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: \((-\infty, -1) \cup (3, \infty)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

For the following exercises, solve the inequality involving absolute value. Write your final answer in interval notation. $$ |2 x+3|<7 $$

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