Real numbers x, y, and z satisfy the inequalities 0 \lt x \lt 1, \quad -1 \lt y \lt 0, \quad and \quad 1 \lt z \lt 2. Which of the following numbers is necessarily positive?

2017 AMC 12B Problem 2 — Solution

2017 AMC 12B Problem 2

Below is the professionally curated solution for Problem 2 of the 2017 AMC 12B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2017 AMC 12B solutions, or check the answer key.

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Concepts:inequality counterexample

Difficulty rating: 1020

Real numbers $x,$ $y,$ and $z$ satisfy the inequalities

$0 \lt x \lt 1, \quad -1 \lt y \lt 0, \quad \text{and} \quad 1 \lt z \lt 2.$

Which of the following numbers is necessarily positive?

$y + x^2$

$y + xz$

$y + y^2$

$y + 2y^2$

$y + z$

Solution:

Adding $y \gt -1$ and $z \gt 1$ gives $y + z \gt 0,$ so $y + z$ is always positive. Each of the other four choices can be made negative: with $x = \tfrac18,$ $y = -\tfrac14,$ $z = \tfrac32,$ every one of $y + x^2,$ $y + xz,$ $y + y^2,$ and $y + 2y^2$ is negative.

Thus, the correct answer is E.

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