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Chapter 3: Problem 46

Solve and write interval notation for the solution set. Then graph the solution set. $$|x+6| \leq 10$$

Short Answer

Expert verified
The solution in interval notation is \([-16, 4]\). Graphically, it's a shaded region between -16 and 4 including both endpoints.

Step by step solution

01

- Understand the absolute value inequality

The given inequality is \(|x+6| \leq 10\). The absolute value \(|A| \leq B\) implies \(-B \leq A \leq B\).
02

- Write the compound inequality

Using the property of absolute values, convert the inequality \(|x+6| \leq 10\) to a compound inequality: \(-10 \leq x+6 \leq 10\).
03

- Solve the compound inequality

Solve the compound inequality step-by-step. First, subtract 6 from all parts: \(-10 - 6 \leq x+6 - 6 \leq 10 - 6\). This simplifies to \(-16 \leq x \leq 4\).
04

- Write the solution in interval notation

The solution set for the inequality is \(-16 \leq x \leq 4\). In interval notation, this is written as \([-16, 4]\).
05

- Graph the solution set

To graph the solution set \([-16, 4]\), draw a number line with a closed circle at -16 and a closed circle at 4, and shade the region in between to indicate that all numbers between -16 and 4, including -16 and 4, satisfy the inequality.

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Key Concepts

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

compound inequality
A compound inequality involves two separate inequalities joined together by either 'and' (intersection) or 'or' (union). For the given exercise, we start with the absolute value inequality \(|x+6| \leq 10\). To transform this, we use the property of absolute values which tells us that \(|A| \leq B\) can be rewritten as \-B \leq A \leq B\. Using this property, our inequality \(|x+6| \leq 10\) translates to the compound inequality \-10 \leq x+6 \leq 10\. This combined statement means that the variable \(x\) must satisfy both conditions at the same time, lying between \-10\ and \10\ (inclusive). We further simplify this by isolating \(x\) and solving for it.
interval notation
Interval notation is a way of writing subsets of the real number line. When using interval notation to describe a solution set, we use square brackets \([\,]\) or parentheses \(\,)\) to indicate whether endpoints are included or excluded. A closed interval, using square brackets, includes its endpoints. In the given exercise, we found that the solution to the inequality \(-10 \leq x+6 \leq 10\) simplifies to \-16 \leq x \leq 4\. In interval notation, this solution is written as \([-16, 4]\), indicating that \-16\ and \4\ are both included in the solution set.
solution set
The solution set is a collection of all values that satisfy a given inequality or equation. For the given inequality \(|x+6| \leq 10\), the solution set includes all \(x\) values between \-16\ and \4\ (inclusive). This set is written as \([-16, 4]\) in interval notation. To visualize the solution set, we use a number line. On this number line, a closed circle is placed at \-16\ and another at \4\, with the region between these two points shaded. This shading indicates that every number between \-16\ and \4\, including these endpoints, is part of the solution set.

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