这部分需要会什么？
- 将绝对值理解为到某点的距离
- 按右侧正负拆分支求解绝对值方程
- 用距离区间表示绝对值不等式
核心概念与公式
\(|x-a|\) 表示 \(x\) 与 \(a\) 的距离。\(|x-a|=d\) 在 \(d>0\) 时有 \(x=a\pm d\)，\(d=0\) 时只有 \(x=a\)，\(d<0\) 时无解。
\(|x-a|\le d\)（\(d\ge0\)）表示 \(a-d\le x\le a+d\)；\(|x-a|\ge d\) 表示 \(x\le a-d\) 或 \(x\ge a+d\)。先把绝对值单独移到一边，再判断另一边的符号。
做题时怎样使用
- 先把绝对值整体隔离到等号或不等号一边
- 判断另一边的正负，再拆成对应的分支或距离区间
- 把候选解代回原式检查，确认分支条件成立
容易混淆的地方
绝对值方程有时只有一个解或无解，不能一律说有两个。绝对值外还带系数时不要直接拆分支，先化成 \(|…|=d\) 的形式。
用一个例题串起方法
知识点教学例题解 \(|2x-3|=5\)。
展开解题思路 ＋
分成 \(2x-3=5\) 或 \(2x-3=-5\)。因此 \(x=4\) 或 \(x=-1\)。两者代回绝对值均为 5。
本知识点题库
先独立作答，再查看答案与解析。
绝对值方程与不等式基础
|-7x +17| =73
What is the sum of the solutions to the given equation?
原库参考解析
绝对值方程与不等式基础
The function f is defined by\( f\left(x\right)=\frac{\left|x\right|}{a}-17, \) where a < 0. What is the product of f(18a) and f(8a)?
原库参考解析
绝对值方程与不等式基础
f(x) = |57-2x|
The function f is defined by the given equation. For which of the following values of k does f(k) = 3k?
原库参考解析
绝对值方程与不等式挑战
The equation 4|x -6|= k, where k is a constant, has exactly one solution. which of the following could be the value of \( \frac{k}{4} \) ?
原库参考解析
绝对值方程与不等式进阶
f (x) = |17 − 2x|
The function f is defined by the given equation. For which of the following values of k does f (k) = 3k ?
原库参考解析
绝对值方程与不等式挑战
The function\( f\left(x\right)=\frac{\left|x\right|}{s}-10 \) where \( s \) < 0. What is the sum of \( f\left(15s\right) \) and\( f\left(3s\right) \)?
原库参考解析
绝对值方程与不等式挑战
f(x)=|71-2x|The function f is defined by the given equation. For which of the following values of k does f(k)=3k?
原库参考解析
绝对值方程与不等式挑战
f (x) = | 71 - 2x |
The function f is defined by the given equation. For which of the following values of k does f(k) = 3k ?
原库参考解析
绝对值方程与不等式挑战
The equation \( 2|x-5|=k, \)where k is a constant, has exactly one solution. Which of the following could be the value of\( \frac{k}{2} \) ?
原库参考解析
绝对值方程与不等式进阶
\( |x-9|=-|x-9|+10 \)
What is the sum of the solutions to the given equation?
原库参考解析
绝对值方程与不等式挑战
The function q is defined by \( q\left(x\right)=a\left|x-9\right|^2-87\left|x-9\right|+b \) , where a and b are constants, and a > b> 87. If q(439) = h and q(-421) = k , where h and k are constants, what is the value of\( 9\left(-439\right)^{h-k}+421\left(9\right)^{k-h} \) ?
原库参考解析
绝对值方程与不等式进阶
\( \left|x-6\right|=15 \)
What is the sum of the solutions to the given equation?
原库参考解析
