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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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