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Chapter 8: Problem 63

Solve and graph. $$ |a| \leq 9 $$

Short Answer

Expert verified
-9 \leq a \leq 9

Step by step solution

01

Understanding Absolute Value Inequality

The given inequality is \( |a| \leq 9 \). Remember that \( |a| \) represents the distance from \(a\) to zero on a number line. Thus, \( |a| \leq 9 \) means that \(a\) is within 9 units from zero.
02

Rewrite the Inequality without Absolute Value

Since the absolute value of a number \(a\) must be less than or equal to 9, this can be written as a compound inequality: \( -9 \leq a \leq 9 \).
03

Graph the Solution

To graph the inequality \( -9 \leq a \leq 9 \), draw a number line and shade the region between \(-9\) and \(9\). Use closed circles at \(-9\) and \(9\) to indicate that these endpoints are included in the solution.

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Key Concepts

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

absolute value
Absolute value refers to the distance of a number from zero on a number line, regardless of direction. In mathematical terms, the absolute value of a number \(a\) is denoted as \(|a|\). This means that \(|a|\) is always non-negative. For instance, \(|5| = 5\) and \(|-5| = 5\). The crucial aspect is that it measures 'how far' the number is from zero but not 'in which direction'. In the context of inequalities, understanding the absolute value is key, as it determines the range of possible solutions.
compound inequalities
Compound inequalities involve combining two simple inequalities. They are used frequently in solving absolute value problems. The inequality \(|a| \leq 9\) means that the value of \(a\) lies between \(-9\) and \(9\). To express this without absolute values, we rewrite it as a compound inequality: \-9 \leq a \leq 9. This tells us that the number \(a\) is greater than or equal to \(-9\) and less than or equal to \(9\) at the same time. Compound inequalities simplify absolute value expressions by breaking them down into more manageable parts.
graphing inequalities
Graphing inequalities helps visualize the solutions on a number line. To graph \(-9 \leq a \leq 9\), follow these steps:
  • Draw a horizontal line (a number line).
  • Mark the points \(-9\) and \(9\) on the line.
  • Shade the entire region between \(-9\) and \(9\) to indicate that all numbers in this interval are solutions.
  • Use closed circles at the points \(-9\) and \(9\) to show that these boundaries are included in the solution.
This visual representation helps in clearly understanding the range of values that satisfy the inequality.

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Most popular questions from this chapter

The function given by \(w(t)=0.0125 t+4.525\) can be used to estimate the number of pounds of solid waste, \(w(t),\) produced daily, on average, by each person in the United States, \(t\) years after \(2000 .\) For what years will waste production range from 4.6 lb to 4.8 lb per person per day?

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?

Counseling. The function given by \(s(t)=500 t+16,500\) can be used to estimate the number of student visits to Cornell University's counseling center \(t\) years after 2000 . For what years is the number of student visits between \(18,000\) and \(21,000 ?\)

Use a graphing calculator to graph each function and find solutions of \(f(x)=0 .\) Then solve the inequalities \(f(x)<0\) and \(f(x)>0\). $$ f(x)=\frac{x^{3}-x^{2}-2 x}{x^{2}+x-6} $$

Solve. $$ x^{2}+x-2<0 $$
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