AMC12 2017 B

AMC12 2017 B · Q2

AMC12 2017 B · Q2. It mainly tests Linear inequalities.

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

实数 $x, y,$ 和 $z$ 满足不等式 $0 < x < 1, \quad -1 < y < 0, \quad \text{and} \quad 1 < z < 2$。以下哪个数一定是正的？

(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 $y + z > 0$。其他四个选项在例如 $x = \frac{1}{8}$，$y = -\frac{1}{4}$，$z = \frac{3}{2}$ 时值为负。

Topics

AL.003 Linear inequalities