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Chapter 1: Problem 73

Solve the absolute value inequality. Express the answer using interval notation and graph the solution set. $$|x| \leq 4$$

Short Answer

Expert verified
The solution is \([-4, 4]\).

Step by step solution

01

Understanding Absolute Value Inequality

The inequality \(|x| \leq 4\) involves the absolute value of \(x\). This means that \(x\) is at most 4 units away from zero on the number line. The absolute value inequality \(|x| \leq 4\) implies both \(-4 \leq x \) and \(x \leq 4\).
02

Break Down the Absolute Value Inequality

Rewrite the absolute value inequality \(|x| \leq 4\) as a compound inequality: \(-4 \leq x \leq 4\). This shows the range of values that \(x\) can take, satisfying the inequality.
03

Express in Interval Notation

Convert the compound inequality \(-4 \leq x \leq 4\) into interval notation. The solution in interval notation is \([-4, 4]\), which includes all real numbers from -4 to 4, inclusive.
04

Graph the Solution Set

To graph the solution set, draw a number line, and shade the region between -4 and 4. Use closed circles or dots at -4 and 4 to show that these boundary points are included in the solution set, representing the interval \([-4, 4]\).

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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 mathematical tool used to express a range of values in a compact form. When we deal with inequalities, like \-4 \leq x \leq 4\, interval notation helps us describe all possible values that \(x\) can take, within a defined range, by using brackets.
  • Square brackets \([-4, 4]\) indicate that the endpoints -4 and 4 are included in the range. This happens when the inequality symbol is "less than or equal to" or "greater than or equal to."
  • Parentheses, on the other hand, suggest that an endpoint is not included, which is common with strict inequalities ("less than" or "greater than", e.g., \(( -4, 4 )\)).
Interval notation is incredibly useful in mathematics as it provides a clear and concise way to express a set of solutions. This notation ensures that anyone looking at the interval immediately understands which numbers are included and which are not.
Compound Inequalities
Compound inequalities involve two separate inequalities that are combined into a single statement by the word 'and' or 'or'. For example, the inequality \-4 \leq x \leq 4\ is a compound inequality stating that \(x\) must satisfy both \-4 \leq x\ and \x \leq 4\.
Bringing together multiple inequalities allows us to define a range of values that solves both parts simultaneously:
  • If connected by 'and', the solution must satisfy both inequalities at the same time.
  • If linked by 'or', the solution can satisfy either one of the inequalities or both.
In our case of \(|x| \leq 4\), the compound inequality is solved by noting that \(x\) can be as small as -4 and as large as 4, encompassing all values between them. Understanding compound inequalities is key to defining the solution set of absolute value inequalities accurately.
Graphing Inequalities
Graphing inequalities visually represents the solution set on a number line. To graph the solution of \(|x| \leq 4\), you need to show all possible values of \(x\) that fit within the range indicated by the inequality.
  • Draw a number line.
  • Highlight the relevant interval: Shade the number line between -4 and 4, ensuring the entire region is marked to signify all numbers in the interval are solutions.
  • Indicate included endpoints by placing closed circles or dots at -4 and 4. This underscores the fact these points are included in the solution set.
Graphing inequalities is a powerful way to help understand and visualize the range of solutions. It not only aids in comprehension but also provides a quick check that ensures all included values satisfy the original inequality.

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