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Chapter 14: Problem 16

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

Short Answer

Expert verified
Possible values of \(x\) are -3.

Step by step solution

01

Express y and z in terms of x

Because \(x, y,\) and \(z\) are given to be consecutive odd integers, it's possible to express \(y\) and \(z\) in terms of \(x\). If \(x\) is any odd integer, then the next odd integer (and therefore, \(y\)) would be \(x + 2\), and the one after that (\(z\)) would be \(x + 4\).
02

Substitute y and z into inequality

Substitute \(y = x + 2\) and \(z = x + 4\) into the given inequality \(x + y + z < z\). This gives \(x + (x + 2) + (x + 4) < x + 4\). Simplify this inequality.
03

Simplify and solve the inequality

Simplify the left-hand side by adding like terms, which provides the inequality \(3x + 6 < x + 4\). Solving this for \(x\), first subtract \(x\) and -4 from both sides to get \(2x + 2 < 0\). Simplify again by dividing throughout by 2 to obtain \(x+1<0\). Solving for \(x\) leads to \(x < -1\).
04

Find the valid values of x

Given that \(x < -1\) and \(x\) is an odd integer, the only valid values are those less than -1. Checking this against the values provided, the correct answers are those that meet the criteria. Note that 0 and 1 are not odd integers.

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