2011 AMC 10A 第 7 题

Question

下列哪个方程没有解？

Which of the following equations does not have a solution?

(x + 7)^2 = 0

|-3x| + 5 = 0

\sqrt{-x} - 2 = 0

\sqrt{x} - 8 = 0

|-3x| - 4 = 0

找出哪个方程要求一个非负表达式等于负数。

Check which equation asks a nonnegative expression to be negative

绝对值加

5

不可能等于

0

。

An absolute value plus

5

cannot equal

0

选项 A 化为

x + 7 = 0

x = -7\text{，}

因此有解。

选项 B 化为

|-3x| = -5\text{，}

绝对值不可能为负数，因此无解。

其他选项都有解，如下面计算所示。

选项 C 化为

\sqrt{-x} = 2

-x = 4

x = -4\text{，}

所以有解。

选项 D 化为

\sqrt{x} = 8

x = 64\text{，}

所以也有解。

最后，选项 E 化为

|-3x| = 4

-3x = \pm 4

x = \pm \dfrac{4}{3}\text{，}

同样有解。

所以正确答案是 B。

A simplifies to

x + 7 = 0

x = -7,

so it has a solution.

B simplifies to

|-3x| = -5,

which has no solution since absolute value makes everything positive.

Let us make sure that all the other choices have solutions.

C simplifies to

\sqrt{-x} = 2

-x = 4

x = -4,

which is fine.

D simplifies to

\sqrt{x} = 8

x = 64,

which works.

Finally, E simplifies to

|-3x| = 4

-3x = \pm 4

x = \pm \dfrac{4}{3},

which has a solution as well.

Thus, B is the correct answer.

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