Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 2: Problem 33
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 set is \((-
fty, -1) \cup (3,
fty)\).
Step by step solution
01
Understanding the Inequality
The inequality given is \(|x-1|>2\). This means we need to find all \(x\) values for which the distance between \(x\) and 1 is greater than 2. We will split this into two separate inequalities.
02
Breaking Down the Absolute Value Inequality
We can split the inequality \(|x-1|>2\) into two separate inequalities: \(x-1 > 2\) and \(x-1 < -2\). We will solve these inequalities separately.
03
Solve First Inequality
For \(x-1 > 2\), adding 1 to both sides gives us \(x > 3\). This means \(x\) must be greater than 3.
04
Solve Second Inequality
For \(x-1 < -2\), adding 1 to both sides gives us \(x < -1\). This means \(x\) must be less than -1.
05
Combine the Solutions
The solution for the inequality \(|x-1|>2\) is the union of the solutions from the two inequalities: \(x < -1\) or \(x > 3\). We can express this in interval notation as \((-fty, -1) \cup (3, fty)\).
06
Graph the Solution
On a number line, we shade to the left of -1 and to the right of 3, indicating the regions where the inequality holds true. The regions \((-fty, -1)\) and \((3, fty)\) do not include -1 and 3 themselves, as indicated by open circles at -1 and 3.
07
Final Representation and Interval Notation
The solution set on the number line corresponds to values of \(x\) outside the interval \([-1, 3]\). The solution to the inequality \(|x-1|>2\) in interval notation is \((-fty, -1) \cup (3, fty)\).
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Graphing Inequalities
When it comes to graphing inequalities on a number line, the main goal is to visually represent the solution set of the inequality. For an inequality like \(|x-1|>2\), we need to show where the inequality holds true. This involves identifying on the number line:
- The regions that satisfy the inequality.
- Whether certain points are included or excluded.
- Shade all numbers less than -1.
- Leave -1 itself unshaded, indicating it's not part of the solution (open circle).
- Also shade numbers greater than 3.
- Leave 3 itself unshaded (open circle as well).
Interval Notation
Interval notation is a concise way to describe sets of numbers on a number line. It's especially handy for inequalities because it clearly shows which numbers are included or excluded from the solution set. For instance, let's take the solution to \(|x-1|>2\): \(x < -1\) or \(x > 3\).
- This translates to two separate intervals: \((-\infty, -1)\) and \((3, \infty)\).
- Parentheses \(()\) are used because -1 and 3 are not included in the solution, which matches the open circles on the graph.
- The union symbol \(\cup\) is used to combine the two intervals, indicating the complete solution set.
Solution Sets
A solution set is simply the collection of numbers that satisfy a given inequality. When dealing with inequalities involving absolute values, the solution set often comprises multiple intervals. For our example \(|x-1|>2\), the solution set includes:
- All \(x\) values less than -1.
- All \(x\) values greater than 3.
Inequality Solving Steps
Solving absolute value inequalities involves a structured approach to ensure all solutions are found. Here's how it's done:
- Step 1: Understand the Inequality: Absolute value inequalities like \(|x-1|>2\) measure distance from a point on a number line. We're looking for points more than 2 units away from 1.
- Step 2: Split the Inequality: The inequality is divided into two: \(x - 1 > 2\) and \(x - 1 < -2\). This comes from the nature of absolute values needing to consider both the positive and negative distances.
- Step 3: Solve Each Part Separately: Solve \(x - 1 > 2\) by adding 1 to find \(x > 3\). Solve \(x - 1 < -2\) similarly to find \(x < -1\).
- Step 4: Combine the Solutions: Merge the separate solutions using union, resulting in \(x < -1\) or \(x > 3\), and express in interval notation: \((-\infty, -1) \cup (3, \infty)\).
