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Chapter 2: Problem 32

For Problems \(31-54\), solve each inequality. $$ |x-3|>9 $$

Short Answer

Expert verified
\(x < -6\) or \(x > 12\).

Step by step solution

01

Understanding the Absolute Value Inequality

The expression \(|x-3|>9\) means you want to find all values of \(x\) that make the distance between \(x\) and 3 greater than 9. This means we need to consider two separate cases: \(x-3 > 9\) or \(x-3 < -9\).
02

Case 1: Solving \(x-3>9\)

To solve \(x-3>9\), we add 3 to both sides of the inequality to isolate \(x\). This gives us \(x>12\).
03

Case 2: Solving \(x-3

To solve \(x-3<-9\), we again add 3 to both sides. This time, it yields \(x<-6\).
04

Combining the Solutions

The solution to the inequality \(|x-3|>9\) is the union of the solutions from both cases. Thus, \(x< -6\) or \(x> 12\). In interval notation, this solution is written as \((-鈭�, -6) \cup (12, 鈭�)\).

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Key Concepts

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

Solving Absolute Value Inequalities
When facing an absolute value inequality like \(|x-3|>9\), remember that absolute value represents distance. Here, it means finding numbers whose distance from 3 is greater than 9. To break it down:
  • The absolute value inequality \(|x-3|>9\) suggests two scenarios: the number is more than 9 units away to the left or to the right of 3 on a number line.
  • We split this inequality into two distinct simpler inequalities to solve: \(x-3 > 9\) and \(x-3 < -9\).
Let's solve each:
  • For the inequality \(x-3>9\), add 3 to both sides to get \(x>12\).
  • For \(x-3<-9\), also add 3 to both sides to get \(x<-6\).
These solutions show us where \(x\) lies on a number line, ensuring the original condition is met. The key is taking apart the absolute value inequality into manageable pieces.
Understanding Interval Notation
Interval notation offers a neat way to describe the set of solutions for an inequality. Let's delve deeper:
  • After solving the absolute value inequality \(|x-3|>9\), we found two conditions: \(x>12\) and \(x<-6\).
  • These conditions mean \(x\) is either less than \(-6\) or greater than \(12\). Neither includes \(-6\) or \(12\) because \(>\) and \(<\) are strict inequalities.
To write this in interval notation:
  • \((12, 鈭�)\) represents numbers greater than 12 moving towards infinity, excluding 12 itself.
  • \((-鈭�, -6)\) indicates numbers falling below \(-6\) extending to negative infinity, excluding \(-6\).
Put together, use the union symbol \(\cup\) to express the full solution set: \((-鈭�, -6) \cup (12, 鈭�)\). This notation is crucial as it concisely shows the entire range of possible solutions in one line.
Inequality Solutions and Their Interpretation
Solving absolute value inequalities involves not just calculating the numbers but understanding their meaning and interpreting solutions correctly:
  • Each solved inequality points to a range of numbers that fulfill the original condition.
  • For \(x-3 > 9\), the solution \(x>12\) represents any number larger than 12 satisfying the distance condition from 3.
  • Similarly, \(x-3 < -9\) leading to \(x<-6\) describes numbers less than \(-6\) that work.
The union of these solutions \((-鈭�, -6) \cup (12, 鈭�)\) shows that any value of \(x\) falling outside the interval \([-6, 12]\) is valid. Visualizing these sets can further aid understanding:
  • The number line approach works wonders here by marking regions like \((-鈭�, -6)\) and \((12, 鈭�)\), highlighting the portions of interest.
  • Graphs can clearly showcase where \(x\) meets the inequality's criteria, giving a visual affirmation of the solution set.
Mastering these methods ensures you not only solve inequalities but interpret them with finesse, turning numbers into understandable conditions about quantities.

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Most popular questions from this chapter

For Problems \(35-44\), solve each compound inequality and graph the solution sets. Express the solution sets in interval notation. $$ x-2>-1 \quad \text { and } \quad x-2<1 $$

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For Problems \(17-30\), solve each inequality and graph the solution. $$ |x|>3 $$

Solve each inequality. $$ |7 x-6| \geq 22 $$

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