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Chapter 2: Problem 52

Solve and graph the solution set. In addition, give the solution set in interval notation. $$|x| \leq 2$$

Short Answer

Expert verified
The solution is \([-2, 2]\), graphed as a closed interval between -2 and 2.

Step by step solution

01

Understand the Absolute Value Inequality

The inequality given is \(|x| \leq 2\). This means that the distance of \(x\) from 0 is at most 2 units. We need to break this absolute value inequality into two separate inequalities without the absolute value.
02

Break Down the Inequality

The inequality \(|x| \leq 2\) means that \(-2 \leq x \leq 2\). This is because the absolute value notation captures both the positive and negative scenarios, and \(x\) cannot be less than -2, nor can it be greater than 2.
03

Graph the Solution Set

Draw a number line. Mark the values -2 and 2. Since the inequality is \(\leq\) and not \(<\), use closed circles at -2 and 2 to include these endpoints in the solution set. Shade the region between -2 and 2 to show all numbers \(x\) that satisfy \(-2 \leq x \leq 2\).
04

Express the Solution in Interval Notation

For the interval notation, we use brackets \([\ldots]\) to indicate that endpoints are included. Therefore, the solution set in interval notation is \([-2, 2]\).

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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 of expressing an interval on the number line. It tells us which numbers are included in a set. When dealing with absolute value inequalities like \(|x| \leq 2\), once we break it down into \-2 \leq x \leq 2\, we can write the solution set using interval notation.
  • Brackets: The square brackets \[\] means that the endpoints are included in the set.
  • Example: The interval notation for this inequality is \([-2, 2]\).
Whenever the endpoints of an interval are part of the solution, always use brackets. This differs from using parentheses, which indicate that the endpoints are not included.
Number Line
A number line is a straight horizontal line that visually represents numbers. It helps in understanding inequalities and graphing them. When using it to graph \(-2 \leq x \leq 2\), here's how you do it:
  • Draw a straight line and mark several points to represent numbers. Make sure to include the critical points, -2 and 2.
  • Use closed circles at -2 and 2 because the inequality \(\leq\) includes these endpoints.
  • Shade the region between -2 and 2 to show all possible solutions.
The number line provides a visual way to see the range of numbers that satisfy the inequality, making it easier to understand the solution set.
Solution Set
The solution set of an inequality represents all the numbers that satisfy the inequality condition. In the case of \(|x| \leq 2\), the solution set is all numbers between -2 and 2, inclusive.
  • This means any number \(-2 \leq x \leq 2\) is part of the solution set.
  • Include the endpoints \-2 and 2 since the inequality is \(\leq\).
Expressing the solution in different ways, such as graphing on a number line or using interval notation, helps to better understand and communicate the solution set.
Inequality Graphing
Graphing inequalities on a number line is a straightforward way to visualize solutions. For the inequality \(-2 \leq x \leq 2\), graphing it involves these steps:
  • Draw a number line that includes the range of interest.
  • At each endpoint, use closed circles because \(\leq\) indicates inclusion of the endpoints.
  • Shade the region between -2 and 2 to represent the solution set visually.
Graphing allows us to see quickly which values of \(x\) satisfy the inequality and communicate these visually in a simple way.

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