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Chapter 3: Problem 40

Solve and write interval notation for the solution set. Then graph the solution set. $$|x| \geq 2$$

Short Answer

Expert verified
The solution set in interval notation is \((-\infty, -2] \cup [2, \infty)\). Graph it with closed dots at -2 and 2, shading left from -2 and right from 2.

Step by step solution

01

Understand the Absolute Value Inequality

The inequality \(|x| \geq 2\) means that the distance of \(x\) from 0 is greater than or equal to 2. This gives us two separate inequalities to solve: \(x \geq 2\) and \(x \leq -2\).
02

Solve the Inequality \(x \geq 2\)

The first part of the inequality is already simple: \(x \geq 2\). This inequality means that \(x\) can be any value starting from 2 and increasing to infinity.
03

Solve the Inequality \(x \leq -2\)

The second part of the inequality states: \(x \leq -2\). This inequality means that \(x\) can be any value starting from -2 and decreasing to negative infinity.
04

Combine the Solutions

Combining the solutions from Step 2 and Step 3, the solution set for the inequality \(|x| \geq 2\) is \(x \in (-\infty, -2] \cup [2, \infty)\).
05

Write the Solution in Interval Notation

The interval notation for the solution is \((-\infty, -2] \cup [2, \infty)\). This means \((-\infty, -2]\) union \([2, \infty)\).
06

Graph the Solution Set

To graph the solution set on a number line, place a closed dot at -2 and shade to the left to negative infinity. Then, place a closed dot at 2 and shade to the right to positive infinity.

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Key Concepts

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

absolute value
The absolute value concept measures the distance a number is from zero on a number line. Absolute value is always positive because distance is never negative. For any real number, say \(x\), the absolute value is denoted as \(|x|\). Here are some key points about absolute value:
  • \(|x| = x\) if \(x \geq 0\) (non-negative numbers)
  • \(|x| = -x\) if \(x < 0\) (negative numbers).
Examples: \(|3| = 3\), because 3 is positive and \(|-3| = 3\), because the distance from -3 to 0 is 3 units. For the inequality |x| \geq 2, x can take on values on either side of the number line, as long as the distance from 0 is greater than or equal to 2.
inequalities
Inequalities are statements that show the relative size or order of two values. In our case, the inequality \( |x| \geq 2 \) means x must be at least 2 units away from zero.
  • \(|x| \geq 2\) is split into \(x \geq 2\) and \(x \leq -2\), because absolute value measures distance from zero.
Inequalities have different symbols that represent the relationship between the values:
  • \(>\): Greater than
  • \(<\): Less than
  • \(eq\): Not equal to
In the context of absolute value inequalities, you often need to consider both directions, resulting in compound inequalities like shown.
interval notation
Interval notation is a way to describe sets of numbers along a number line. It uses parentheses and brackets to show where the set starts and ends.
  • Parentheses \( ( ) \) indicate that the endpoint is not included, meaning the value is less than or greater than but not equal.

  • Brackets \( [ ] \) indicate the endpoint is included, meaning it is less than or equal to or greater than or equal to.
For example, \((- \infty, -2] \cup [2, + \infty) \) means all numbers less than or equal to -2 and all numbers greater than or equal to 2. The union symbol \(\bigcup \) combines two intervals. Therefore, this notation covers both parts of the solution to \( |x| \geq 2 \).
graphing inequalities
Graphing inequalities on a number line helps visualize solutions. To graph \(|x| \geq 2\), follow these steps:
  • Mark points \(-2\) and \2\ on the number line.
  • Place closed dots at these points to include them in the solution (indicated by \( \leq \) or \( \geq \)).
  • Shade the line to the left of \(-2\) to show all values less than or equal to \(-2\).
  • Shade the line to the right of \2\ to show all values greater than or equal to \2\.
This graph helps understand the solution set visually. It shows \(- \infty, -2] \cup [2, + \infty) \).

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