AMC10 2017 B

AMC10 2017 B · Q3

AMC10 2017 B · Q3. It mainly tests Linear inequalities, Inequalities (AM-GM etc. basic).

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?

实数$x$、$y$和$z$满足不等式$0 < x < 1$、$-1 < y < 0$和$1 < z < 2$。下面哪个数必然为正？

(A) $y + x^2$ $y + x^2$

(B) $y + xz$ $y + xz$

(D) $y + 2y^2$ $y + 2y^2$

(E) $y + z$ $y + z$

Answer

Correct choice: (E)

正确答案：（E）

Solution

Adding the inequalities $y > -1$ and $z > 1$ yields $y + z > 0$. The other four choices give negative values if, for example, $x = \frac{1}{8}$, $y = -\frac{1}{4}$, and $z = \frac{3}{2}$.

将不等式$y > -1$和$z > 1$相加得到$y + z > 0$。其他四个选项若取$x = \frac{1}{8}$、$y = -\frac{1}{4}$、$z = \frac{3}{2}$则为负值。

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