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Chapter 8: Problem 70

Solve and graph. $$ |x+4| \leq 1 $$

Short Answer

Expert verified
-5 \leq x \leq -3. The graph is a shaded region from -5 to -3, inclusive.

Step by step solution

01

Understand the Absolute Value Inequality

The inequality \(|x+4| \leq 1\) means the distance between \(x+4\) and 0 is less than or equal to 1.
02

Break Down the Absolute Value Inequality

Rewrite the inequality without the absolute value: \(-1 \leq x+4 \leq 1\). This represents the range where \(x+4\) remains within 1 unit from the origin.
03

Solve the Compound Inequality

To solve \(-1 \leq x+4 \leq 1\), subtract 4 from all parts of the inequality: \(-1 - 4 \leq x+4 - 4 \leq 1 - 4\), which simplifies to \(-5 \leq x \leq -3\). This is the solution for \(x\).
04

Graph the Solution on a Number Line

Draw a number line and shade the region between -5 and -3, including the endpoints, as the inequality is \(\leq \) and not \(<\).

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Key Concepts

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

Absolute Value
Understanding absolute value is crucial for solving inequalities like \(|x+4| \leq 1\). Absolute value represents the distance of a number from zero on the number line, regardless of direction. So, for any real number \(a\), the absolute value \(|a|\) is always non-negative.

In our problem, \(|x+4|\) can be interpreted as the distance between \(x+4\) and zero. The inequality \(|x+4| \leq 1\) tells us that this distance is less than or equal to 1.

Therefore, x must be in the range that makes the expression \(x+4\) stay within 1 unit from zero, which leads us to the next concept: compound inequalities.
Compound Inequality
A compound inequality is made up of two or more inequalities joined by 'and' or 'or'. In this problem, once we remove the absolute value, we obtain: \(-1 \leq x+4 \leq 1\).

This represents two inequalities: \(-1 \leq x+4\) and \(x+4 \leq 1\), which must both be true simultaneously (hence, an 'and' combination).

To solve the compound inequality:
  • Subtract 4 from all parts of \(-1 \leq x+4 \leq 1\).
  • We get: \(-1 - 4 \leq x+4 - 4 \leq 1 - 4\).
  • Simplifying, we find: \(-5 \leq x \leq -3\).
Thus, \(x\) must be between -5 and -3, inclusive. This brings us to graphing the solution on a number line.
Number Line Graph
Graphing solutions on a number line helps visualize the range of values that satisfy the inequality.

For \(-5 \leq x \leq -3\):
  • Draw a horizontal line representing the number line.
  • Mark points -5 and -3 on this number line.
  • Shade the region between -5 and -3, including the endpoints.
Including the endpoints means using solid dots or circles at -5 and -3, because the inequality includes 鈥榣ess than or equal to鈥� (\(leq\)), not just 'less than' (\(<\)).

This visual representation clearly shows all possible values of \(x\) that meet the condition of the inequality. Understanding and using these graphs can make sense of solutions to inequalities more straightforward.

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Most popular questions from this chapter

Height of a Thrown Object. The function $$ S(t)=-16 t^{2}+32 t+1920 $$ gives the height \(S,\) in feet, of an object thrown from a cliff that is 1920 ft high. Here \(t\) is the time, in seconds, that the object is in the air. a) For what times does the height exceed 1920 ft? b) For what times is the height less than 640 ft?

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Solve using substitution or elimination. $$ \begin{aligned} &3 x-2 y=4\\\ &5 x-3 y=5 \end{aligned} $$

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For \(g(x)=\frac{2+3 x}{2 x-4},\) find all \(x\) -values for which \(g(x) \geq 0\).
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