2017 AMC 10B Problem 3 — Solution | LIVE by Po-Shen Loh

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2017 AMC 10B Problem 3

Attempt Problem 3 of the 2017 AMC 10B below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2017 AMC 10B solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

3.

Real numbers $x,$ $y,$ and $z$ satisfy the inequalities $0 < x < 1,$ $-1 < y < 0,$ and $1 < z < 2.$

Which of the following numbers is necessarily positive?

$y+x^2$

$y+xz$

$y+y^2$

$y+2y^2$

$y+z$

Answer: E

Concepts:inequality counterexample

Difficulty rating: 960

Solution:

Since $-1 < y$ and $1 < z,$ we can add the inequalities to see that $0 < y+z.$ This naturally proves choice E correct.

Furthermore, we can eliminate every other choice with the following values: $x=0.1,$ $y=-0.25,$ $z=1.25.$

Thus, the correct answer is E .

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