Question

方程的所有解之和是多少？ $x = |2x-|60-2x||?$

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

$32$

$60$

$92$

$120$

$124$

答案：C

知识点：绝对值分类讨论

难度评级：1660

解答：

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

这两个方程分别化为和 $x = |60 - 2x|$ $3x = |60 - 2x|.$

对第一个方程，有 $2$ 个线性方程要解： $x = 60 - 2x \text{ and } x = 2x - 60.$

分别求解得以及 $3x = 60$ $x = 20$ $-x = -60$ $x = 60.$

对第二个方程，有 $3x = 60 - 2x \text{ and } 3x = 2x - 60.$

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

所有解之和为 $20 + 60 + 12 = 92.$

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 $x = 60 - 2x \text{ and } x = 2x - 60.$

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

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

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

解答：

其他年份的第 13 题