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Chapter 0: Problem 3

Find all numbers with absolute value \(9 .\)

Short Answer

Expert verified
The numbers with an absolute value of 9 are 9 and -9. So, the solution is \(\{9, -9\}\).

Step by step solution

01

Understand the absolute value function

The absolute value function is defined as follows: For any number x, \(|x|=\begin{cases} x & x\geq 0 \\ -x & x<0 \\ \end{cases}\) The function returns the distance of a number from 0 on the number line. Positive numbers have the same absolute value as their distance from 0, while negative numbers have the opposite of their distance from 0 as the absolute value.
02

Find the positive number with an absolute value of 9

In this step, we need to find a positive number x such that |x| = 9. Since x is positive, the definition of the absolute value tells us that |x| = x. Thus, we want to find the number x for which x = 9. In this case, x = 9 satisfies this condition, so 9 is the positive number with an absolute value of 9.
03

Find the negative number with an absolute value of 9

In this step, we need to find a negative number x such that |x| = 9. Since x is negative, the definition of the absolute value tells us that |x| = -x. Thus, we want to find the number x for which -x = 9. Multiplying both sides of this equation by -1 gives x = -9. Thus, -9 is the negative number with an absolute value of 9.
04

Combine the results from Steps 2 and 3 to form a solution

We found two numbers with an absolute value of 9: the positive number 9 and the negative number -9. Therefore, the solution is {9, -9}.

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Most popular questions from this chapter

The intersection of two sets of numbers consists of all numbers that are in both sets. If \(A\) and \(B\) are sets, then their intersection is denoted by \(A \cap B .\) In Exercises \(47-\) \(56,\) write each intersection as a single interval. $$(-\infty,-6] \cap(-8,12)$$

The intersection of two sets of numbers consists of all numbers that are in both sets. If \(A\) and \(B\) are sets, then their intersection is denoted by \(A \cap B .\) In Exercises \(47-\) \(56,\) write each intersection as a single interval. $$(-\infty,-3) \cap[-5, \infty)$$

Write each set as an interval or as a union of two intervals. $$\left\\{x:|x+4|<\frac{\varepsilon}{2}\right\\} ; \text { here } \varepsilon>0$$

The intersection of two sets of numbers consists of all numbers that are in both sets. If \(A\) and \(B\) are sets, then their intersection is denoted by \(A \cap B .\) In Exercises \(47-\) \(56,\) write each intersection as a single interval. $$[-8,-3) \cap[-6,-1)$$

Simplify the given expression as much as possible. $$\frac{1}{x-y}\left(\frac{x}{y}-\frac{y}{x}\right)$$
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