Question

下列方程的所有解之和是多少？ $x = |2x-|60-2x||$

What is the sum of all the solutions of the equation below? $x = |2x-|60-2x||$

$32$

$60$

$92$

$120$

$124$

答案：C

知识点：绝对值分类讨论

难度评级：1660

小提示：

按 $60-2x$ 的符号分情况。

Split cases based on the sign of $60-2x$

大提示：

处理内层绝对值后，再解得到的绝对值方程。

After resolving the inner absolute value, solve the resulting absolute-value equations

解答：

先处理外层绝对值。有两种可能： $x = 2x - |60 - 2x|$ 或 $x = -2x + |60 - 2x|\text{。}$

这两个方程分别化为 $x = |60 - 2x|$ 和 $3x = |60 - 2x|\text{。}$

每个方程都有 $2$ 种情况。第一个方程给出 $x = 60 - 2x$ 和 $x = 2x - 60\text{。}$

分别求解得 $3x = 60$ $x = 20$ 以及 $-x = -60$ $x = 60\text{。}$

第二个方程给出 $3x = 60 - 2x$ 和 $3x = 2x - 60\text{。}$

再次分别求解，得 $5x = 60$ $x = 12$ 和 $x = -60$ 。由于原方程右边非负， $x$ 不能为负数。

所有解之和为 $20 + 60 + 12 = 92\text{。}$

所以正确答案是 C。

We first take care of the outer absolute value. We have either $x = 2x - |60 - 2x|$ or $x = -2x + |60 - 2x|.$

These simplify to $x = |60 - 2x|$ and $3x = |60 - 2x|.$

We have $2$ cases for each equation. For the first one, we have $x = 60 - 2x$ and $x = 2x - 60.$

Solving both gives us $3x = 60$ $x = 20$ and $-x = -60$ $x = 60.$

For the other equation, we have $3x = 60 - 2x$ and $3x = 2x - 60.$

Again solving both, we have $5x = 60$ $x = 12$ and $x = -60.$ Note that $x$ cannot be negative since the right side of the original equation is nonnegative.

Adding up all the solutions gives us $20 + 60 + 12 = 92.$

Thus, C is the correct answer.

2010 AMC 10B 第 13 题

答案：C

难度评级：1660

小提示：

大提示：

解答：

其他年份的第 13 题