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Chapter 7: Problem 4

If \(-1

Short Answer

Expert verified
The correct option is the third one: \(-1 < x < 0 < x^2\).

Step by step solution

01

Interpret the Given Information

The first step is to understand that \(x\) lies between 0 and \(-1\), which implies that \(x\) is a negative number with greater magnitude than \(-1\).
02

Check First Option: \(x^2 < -1 < x < 0\)

Now consider the inequality \(x^2<-1\). Since the square of a real number is always nonnegative, \(x^2\) will never be less than \(-1\), hence the first option is incorrect.
03

Check Second Option: \(-1 < x < x^2 < 0\)

Now observe that, within given interval, the inequality \(x < x^2\) always holds true because \(x^2\) will always be a positive number (greater than \(x\) which is negative), and this positive number is less than 0. Therefore, the second option is incorrect.
04

Check Third Option: \(-1 < x < 0 < x^2\)

For this inequality to hold, \(x^2\) should always be positive and greater than 0 in the given interval for \(x\). As we have noted, \(x^2\) will indeed be positive (greater than \(x\)), and hence this inequality holds true for all \(x\) in the given range. So, the third option is correct.

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Most popular questions from this chapter

Which of the following expressions is equivalent to \(\frac{12 !}{15 !} ?\) $$ \begin{aligned} & \text { } \frac{\frac{4 \times 3 \times 2 \times 1}{5 \times 4 \times 3 \times 2 \times 1}}{\frac{1}{15 \times 14 \times 13}} \\ & \frac{4}{5} \end{aligned} $$ I don't know.

In 40 seconds, a mountain biker travels 230 feet. What is her speed in miles per hour? (Note: 1 mile \(=5280\) feet) $$ \begin{array}{r} \frac{230 \times 60 \times 60}{40 \times 5280} \\ \frac{230 \times 5280}{40 \times 60 \times 60} \\ \frac{40 \times 60 \times 60}{230 \times 5280} \\ \text { I don't know. } \end{array} $$

If \(a, b\), and \(c\) are distinct prime numbers, how many factors does abc have?

If \(x<0\), which of the following is true? C \(x^3\) is positive. ( \(x^3\) is negative. \(x^2\) could be either positive or negative. I don't know.

Even integer q is the product of two distinct prime numbers. What can you say about the factors of q?
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