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Chapter 14: Problem 16

If \(x, y,\) and \(z\) are consecutive odd integers where \(x

Short Answer

Expert verified
The possible value of x is -3.

Step by step solution

01

Understand the Problem

Firstly, we need to understand that consecutive odd integers follow a specific pattern. For example, if x is an odd integer, then the next consecutive odd integer (y) will be \(x + 2\) and the next one (z) will be \( y + 2\) or \( x + 4\). So we can represent y as \(x + 2\) and z as \(x + 4\). The problem states that \(x + y + z < z\), when we substitute y and z, we get the equation \(x + x+2 + x+4 < x+4\), which simplifies to \(3x + 6 < x + 4\).
02

Solve the Equation

To solve the equation \(3x + 6 < x + 4\), we first subtract x from both sides to get \(2x + 6 < 4\), then subtract 6 from both sides to get \(2x < -2\). Divide both sides by 2 to find x. The resulting inequality is \(x < -1\).
03

Identify Possible Values of x

From the inequality \(x < -1\), we can see that the possible values of x are all odd integers less than -1. Now we look at the answer choices. The only value that satisfies this condition is -3.
04

Double Check

We double-check our solution against the given inequality: It's given \(x + y + z < z\). When we plug -3 in for x, \(y\) and \(z\) become -1 and 1 respectively. Evaluating our inequality we get \((-3) + (-1) + 1 < 1\), which simplifies to \(-3 < 1\), which is true. So, our answer is correct.

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Most popular questions from this chapter

Quantity \(\mathbf{A}\) The greatest odd factor of 78 Quantity \(\mathbf{B}\) The greatest prime factor of 78 a. Quantity A is greater. b. Quantity B is greater c. The two quantities are equal. d. The relationship cannot be determined from the information given.

What is the greatest distance between two vertices of a rectangular solid with a height of \(5,\) a length of \(12,\) and a volume of \(780 ?\) a. 12 b. \(12 \sqrt{2}\) c. 13 d. \(13 \sqrt{2}\) e. \(13 \sqrt{3}\)

If \(4^{x}=1,024,\) then \(\left(4^{x+1}\right)\left(5^{x-1}\right)=\) a. \(10^{6}\) b. \(\left(5^{4}\right)\left(10^{5}\right)\) c. \(\left(4^{4}\right)\left(10^{5}\right)\) d. \(\left(5^{4}\right)\left(10^{4}\right)\) e. \(\left(4^{4}\right)\left(10^{4}\right)\)

Roberta drove 50 miles in 2 hours. Her rate in miles per hour is equivalent to which of the following proportions? Indicate all such proportions. a. 5 to 20 b. 100 to 4 c. 400 to 16 d. 20 to 500 e. 520 to 20

\(k>0\) \(l>1\) Quantity A \(\frac{1}{\frac{1}{k}+\frac{1}{l}}\) Quantity B \(\frac{k l}{\frac{1}{k}+\frac{1}{l}}\) a. Quantity A is greater. b. Quantity B is greater. c. The two quantities are equal. d. The relationship cannot be determined from the information given.
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