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Chapter 9: Problem 43

Solve and graph the solution set on a number line. $$|x+2| \leq 1$$

Short Answer

Expert verified
The solution to the inequality \(|x+2| \leq 1\) is \(-3 \leq x \leq -1\)

Step by step solution

01

Break down the absolute value inequality into two inequality equations

Any absolute value inequality such as \(|x+2| \leq 1\) can be translated to \((x+2) \leq 1\) and \(-(x+2) \leq 1\).
02

Solve the inequalities

Solving for \(x\) in both inequalities gives: \nFor the first inequality: \(x+2 \leq 1\), we solve for \(x\) by subtracting 2 from both sides to get \(x \leq -1\). \nFor the second inequality: \(-(x+2) \leq 1\), distributing the negative sign gives \(-x-2 \leq 1\). Adding 2 to both sides gives \(-x \leq 3\), and multiplying both sides by -1 (flipping the inequality sign) gives \(x \geq -3\).
03

Find intersection of both solutions

The solution to the original inequality is the intersection of the two individual inequalities. In this case, \(x\) must be both greater than or equal to -3 and less than or equal to -1. This can be written as \(-3 \leq x \leq -1\).
04

Represent the solutions on the number line

On a number line, this would be depicted as a closed circle at -3 and -1, and the line segment between them is shaded to reflect all numbers in between these points inclusive.

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Key Concepts

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

Number Line Representation
Understanding how to represent solutions on a number line is crucial for visualizing the range of values that satisfy an inequality. When dealing with absolute value inequalities, like \(|x+2| \leq 1\), once solved, we often end up with a range or segment on the number line.To illustrate, consider the solution \(-3 \leq x \leq -1\). This solution includes all values of \(x\) between -3 and -1, inclusive of both endpoint values. Here鈥檚 how you represent it on a number line:
  • Locate the numbers -3 and -1 on the number line.
  • Use closed circles (filled dots) on these endpoints because the inequality is non-strict (\

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Most popular questions from this chapter

Rewrite each inequality in the system without absolute value bars. Then graph the rewritten system in rectangular coordinates. \\\ |y| \leq 3\end{array}\right.\end{aligned}$$\left\\{\begin{array}{l}|x| \leq 1 \\ |y| \leq 2\end{array}\right.$

Write each sentence as a linear inequality in two variables. Then graph the inequality. The \(y\) -variable is at least 2 more than the product of \(-3\) and the \(x\) -variable.

Graph the solution set of each system of inequalities or indicate that the system has no solution.. \left\\{\begin{array}{l}3 x+y \leq 6 \\\2 x-y \leq-1 \\\x \geq-2 \\\y \leq 4\end{array}\right.

Solve each inequality using a graphing utility. Graph each of the three parts of the inequality separately in the same viewing rectangle. The solution set consists of all values of \(x\) for which the graph of the linear function in the middle lies between the graphs of the constant functions on the left and the right. $$1 \leq 4 x-7 \leq 3$$

The graphs of solution sets of systems of inequalities involve finding the intersection of the solution sets of nwo or more incqualities. By contrast, in Exercises \(53-54\) you will be graphing the union of the solution sets of two inequalities. Graph the union of \(x-y \geq-1\) and \(5 x-2 y \leq 10\)
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