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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.

Line \(A B\) is tangent to the circle \(C\) at point \(A\). The radius of the circle with center \(C\) is 5 and \(B C=\) \(\frac{10 \sqrt{3}}{3}\). Quantity A The length of line segment \(A B\) Quantity B The length of line segment \(A C\) 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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If \(|2 x-3|+2>7,\) which of the following could be the value of \(x ?\) Indicate all such values. a. -4 b. -3 c. -2 d. -1 e. 0 f. 1 g. 2 h. 3

If three boys and three girls sit in a row on a park bench, and no boy can sit on either end of the bench, how many arrangements of the children on the bench are possible? a. 46,656 b. 38,880 c. 1,256 d. 144 e. 38
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