AMC8 2007

AMC8 2007 · Q15

AMC8 2007 · Q15. It mainly tests Linear inequalities.

Let a, b and c be numbers with 0 < a < b < c. Which of the following is impossible?

设 a、b 和 c 是满足 0 < a < b < c 的数。以下哪个是不可能的？

(A) a + c < b a + c < b

(B) a · b < c a × b < c

(D) a · c < b a × c < b

(E) b/c = a b/c = a

Answer

Correct choice: (A)

正确答案：（A）

Solution

(A) Because $b<c$ and $0<a$, adding corresponding sides of the inequalities gives $b<a+c$, so (A) is impossible. To see that the other choices are possible, consider the following choices for $a$, $b$, and $c$: (B) and (C): $a=1$, $b=2$, and $c=4$; (D): $a=\dfrac{1}{3}$, $b=1$, and $c=2$; (E): $a=\dfrac{1}{2}$, $b=1$, and $c=2$.

（A）因为 $b<c$ 且 $0<a$，将不等式对应边相加可得 $b<a+c$，所以（A）不可能。为了说明其他选项是可能的，考虑下面这些 $a$、$b$、$c$ 的取值：（B）和（C）：$a=1$，$b=2$，$c=4$；（D）：$a=\dfrac{1}{3}$，$b=1$，$c=2$；（E）：$a=\dfrac{1}{2}$，$b=1$，$c=2$。

Topics

AL.003 Linear inequalities