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Chapter 3: Problem 39

Solve and write interval notation for the solution set. Then graph the solution set. $$|x|>3$$

Short Answer

Expert verified
The solution set is \((-\text{鈭瀩, -3) \cup (3, \text{鈭瀩)\).

Step by step solution

01

Understand the Absolute Value Inequality

The inequality given is \(|x| > 3\). This means the distance of \(x\) from 0 on the number line is greater than 3.
02

Split the Inequality

Absolute value inequalities like \(|x| > a\) can be split into two separate inequalities: \(x > a\) and \(-x > a\). For \(|x| > 3\), this splits into two cases: \(x > 3\) or \(x < -3\).
03

Write the Interval Notation

The solutions to these inequalities are \(x > 3\) and \(x < -3\). Writing them in interval notation, the solution set is \((-\text{鈭瀩, -3) \cup (3, \text{鈭瀩)\).
04

Graph the Solution Set

On the number line, shade the regions to the left of \(-3\) and to the right of \(3\). Use open circles at \(-3\) and \(3\) since those points are not included in the solutions.

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Key Concepts

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

Interval Notation
When dealing with inequalities, expressing the solution set clearly is essential. This is where interval notation comes in handy. Interval notation shows the range of values that satisfy an inequality.

For instance, in our example \(|x| > 3\), we split it into two cases: \(x > 3\) or \(x < -3\). The solution to \(x > 3\) includes all numbers greater than 3, which is represented as \( (3, \text{鈭瀩)\). The solution to \(x < -3\) includes all numbers less than -3, written as \((-\text{鈭瀩, -3)\).

Combining both parts, the overall solution set is expressed in interval notation as \( (-\text{鈭瀩, -3) \bcup (3, \text{鈭瀩)\).

Key Points:
  • Use parentheses \( ( \text{\textellipsis } )\) to denote exclusion of endpoints.
  • Use \( \text{\textellipsis } \bcup \text{\textellipsis }\) to combine disjoint intervals.
  • \( \text{鈭瀩 \text{\textellipsis }\) includes all values up to 卤infinity.
Graphing Inequalities
Graphing inequalities helps visualize the solution set on a number line. This complements the interval notation by providing a clear visual understanding.

In our exercise, to graph \(|x| > 3\):
  • Draw a number line.
  • Identify critical points -3 and 3.
  • Place open circles at -3 and 3 to indicate they are not included in the solution.
  • Shade the regions left of -3 and right of 3, representing \( (-\text{鈭瀩, -3) \bcup (3, \text{鈭瀩)\).

A correct graph shows two shaded regions with a gap in between them at -3 and 3, making it clear that these specific points are excluded.

Key Graphing Tips:
  • Use open circles for '>' or '<' inequalities.
  • Use closed circles for '鈮�' or '鈮�' inequalities.
  • Always recheck the shaded regions to ensure they correctly represent the inequality.
Inequality Solutions
Finding solutions to absolute value inequalities involves understanding what the inequality represents.

For example, \( |x| > 3 \) means that the distance of \( x \) from 0 is more than 3 units. To solve it, we break it down as follows:
  • Split into two separate inequalities: \( x > 3 \) and \( x < -3 \).
  • Solve each inequality:
  • For \( x > 3 \): Any \( x\) greater than 3 satisfies this condition.
  • For \( x < -3 \): Any \( x\) less than -3 meets this requirement.

Combining these, we see that the solution set is any number except -3 and 3, forming the intervals \( (-\text{鈭瀩, -3) \bcup (3, \text{鈭瀩)\).

Summary Points:
  • Split the absolute value inequality into two cases.
  • Solve each inequality independently.
  • Combine the solutions using union for a comprehensive set.

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Solve and write interval notation for the solution set. Then graph the solution set. $$|5-2 x|>10$$

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