2017 AMC 12B Problem 2

Problem 2 of 25EasierAlgebra

Real numbers

x,

y,

and

z

satisfy the inequalities

0 \lt x \lt 1, \quad -1 \lt y \lt 0,

and

1 \lt z \lt 2.

Which of the following numbers is necessarily positive?

Show solution

Solution

Adding

y \gt -1

and

z \gt 1

gives

y + z \gt 0,

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.